Getting an Aliyah in the middle of Davening:
A. The law if one was already called up:
If a Kohen is in the midst of reading Kerias Shema [or Birchas Shema], and was called up [by name] to the Torah, some Poskim rule he may not stop [to go up to the Torah for the Aliya]. [Certainly, a Levi or Yisrael may not stop in middle for an Aliya.] Other Poskim, however, rule that even a Yisrael who is in the midst of Kerias Shema, and was called up [by name] to the Sefer Torah, is to stop even in middle of the paragraph. Practically, so is the custom in these provinces [of Ashkenaz, that even if a Yisrael was already called up in middle of Shema, he is to stop and go to the Torah for the Aliyah], and one is not to swerve from the custom, in order to avoid dispute. [According to all, one who is in the middle of Pesukei Dezimra and was called up by name, may stop and go up for the Aliya. According to all, one who already recited Goal Yisrael, and certainly one who is in the midst of Shemoneh Esrei, may not stop for the Aliyah. One who is in the midst of the verse of Shema or Baruch Sheim Kevod Malchuso, must finish until the end of Leolam Vaed, and only then may he go up for the Aliyah. Furthermore, whenever one is called up in middle of a paragraph, he should hurry and try to reach the end of the paragraph, or at the very least, the end of that topic, and only then go up. If, however, doing so will cause a delay to the reading, then he is immediately stop wherever he is holding and go up to the Torah.]
B. Lechatchilah – Initially calling one up:
Yisrael: According to all opinions, a Yisrael who is in the midst of Kerias Shema, or Pesukei Dezimra, may not be called for an Aliyah. [However, he may be called up even initially, between Yishtabach and Yotzer. However, some Poskim rule that this applies even in-between Yishtabach and Yotzer.]
Kohen/Levi-Past Yotzer Or: If a Kohen [or Levi] is in the midst of reading Kerias Shema [or Birchas Shema], one may not initially call him to the Torah [even if he is the only Kohen in Shul], and rather a Yisrael is called up in his place. [One is not to delay until the Kohen finishes Shemoneh Esrei in order for him to be called up.] In such a case, however, it is best for the Kohen to leave the Shul, prior to the Aliyah. [However, even if he did not leave the Shul, the Chazan may say Yisrael Bemikom Kohen.] [However, other Poskim rule, that if there is no other Kohen [or Levi] in Shul then one may even initially call the Kohen to the Aliya. Accordingly, some Poskim conclude that if he is the only available Kohen or Levi in Shul, then one may be lenient to call him up if he is between the paragraphs [of Shema or Birchas Kerias Shema]. Although, it is best for him to leave the Shul prior to the Aliyah and thus not be called up. However, according to Admur, one should not call him up even between the Perakim, as is the simple implied ruling of the Poskim.]
Kohen/Levi-Past Baruch Sheamar: If the Kohen [or Levi] is in the midst of Pesukei Dezimra, then according to all one may even initially call him up if he is the only Kohen or Levi in Shul.
C. Reading along in the Torah:
In the event that one was called up to the Torah in the midst of reading [the blessings of Shema or the paragraphs of] Shema, he is not to read at all inside the Torah together with the Baal Korei. [If, however, he is in the middle of Pesukei Dezimra, some Poskim rule he may read along with the Baal Korei. Other Poskim, however, rule he may not read along even when in the midst of Pesukei Dezimra.]
D. Mi Shebeirach:
One who received an Aliyah in middle of reading the Shema [or Pesukei Dezimra] is certainly not to stop to request the Chazan/Gabaiy to recite a Mi Shebeirach [on his behalf]. [If, however, the Gabay already began the Mi Shebeirach and then forgot his name, he may say his name.]
E. Resuming the prayers:
After one completes the Aliyah he is to resume his reading of Shema [or other part of Davening] from the place he left off. He is not required to return to the beginning of Shema even if the interval was of enough time to have read the entire Shema.
Lechatchila: It is initially forbidden for the Gabaiy to call any person up for an Aliyah if he is in the middle of Pesukei Dezimra, or Birchas Shema [past Yotzer Or], unless he is the only Kohen or Levi in Shul, and is within Pesukei Dezimra [and not past Yotzer Or]. [Some Poskim however allow one who is between Yishtabach and Yotzer Or to be called up even initially.] When a Kohen or Levi is past Yotzer Or and thus cannot be called up, he is to leave Shul prior to the Aliyah, if he is the only Kohen or Levi present.
Bedieved: If one who is past Baruch Sheamar, but before Goal Yisrael, was called up to the Torah, then according to Ashkenazi custom he is to go up for the Aliyah and say the blessings. This applies even if he is a Yisrael, and even if he was in the midst of reading the Shema. If the person was called up in middle of a paragraph, he should hurry and try to reach the end of the paragraph, or end of that topic, if it will not delay the reading. If he is past Yotzer Or, he is not to read at all inside the Torah together with the Baal Korei. If, however, he is in the middle of Pesukei Dezimra, he may read along with the Baal Korei.] He certainly may not stop to request the Chazan/Gabaiy to recite a Mi Shebeirach. After one completes the Aliyah, he is to resume his Davening from the place he left off.
If one who is past Baruch Sheamar was motioned up for an Aliyah, or called by his first name, but his full name was not said, is he to go up and may he say his full name?
It is questionable if he is to go up, even if his first or last name was mentioned aloud, so long as his full name and father’s name which he usually uses to get an Aliya was not mentioned. Certainly, he may not go up if he was only privately asked or motioned. If, however, the Gabai is persistent, then one may do so due to respect of the congregation, but is not to say his name. If, however, the Gabaiy is also persistent about his name, then he may say it.
If one wants an Aliyah, and he knows he will be past Baruch Sheamar, may he ask the Gabaiy beforehand to call him up?
No, as stated above that initially he may not stop in middle for an Aliya. However, if he will be between Yishtabach and Yotzer Or, then some Poskim are lenient, as stated above.
May one who is past Baruch Sheamar purchase an Aliya?
No. This applies even if he will not need to verbalize the amount of money he is offering. However, if he is between Yishtabach and Yotzer Or, then some Poskim are lenient, as stated above.
If one who is past Baruch Sheamar was called up for Maftir, is he to also read the Haftorah?
If he is in the midst of Pesukei Dezimra, he is to read the blessings of the Haftorah and the Haftorah. If he is in the midst of Birchas Shema or Shema, then if it is accepted in that community for one person to say the blessings and have another say the Haftorah, then he is to say the blessings of the Haftorah and have another person read the Haftorah for him. However, in those communities that the custom is always for the Maftir to read the Haftorah, then he is to do so.
May one be a Baal Korei to read from the Torah if he is in the middle of Davening?
If the Sefer Torah has been removed and the only available Baal Korei is in middle of Davening [Baruch Sheamar until Shemoneh Esrei], he may stop and read from the Torah. This applies even if he is in middle of Shema, or Birchas Kerias Shema, although initially he is to try to finish until the end of the paragraph. In such a case, he is not to be the Gabai who calls people to the Torah.
May a lone Kohen who is in the midst of Pesukei Dezimra say his name to be called up for the Torah reading?
It is best for him to go up without mentioning his name. If, however, the Gabaiy asks for it, then he may say it.
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 Admur 66/6
 Olas Tamid 66; M”B 66:24; Kaf Hachaim 66:24
 Kneses Hagedola 66; Kesher Gudal 11:22; Kaf Hachaim 66:24
 1st opinion in Admur ibid; 2nd opinion in Michaber 66:4; Stam opinion in Michaber 135:5; Opinion in Beis Yosef 135; Rashba 1:185; See Kneses Hagedola 66 who wants to say the Rashba and Manhig do not argue, and the Rashba is talking of Lechtachila, while the Manhig of Bedieved; See however Peri Chadash 66:4 and Kaf Hachaim 66:24 who negates this explanation
 The reason: As the Sages [Gittin 59b] only required one to precede a Kohen, due to the honor of his priesthood, as if he is available and another person is called, it ends up that we are being lax in the honor of his priesthood. However, when the Kohen is not called up due to being in the midst of Kerias Shema, there is no belittlement to the honor of his priesthood in the fact that a Yisrael goes up in his place. [Admur ibid] Likewise, being that he is unable to get the Aliya for Halachic reasons, his refusal is not included in the Talmudic warning regarding one who refuses an Aliyah. [Kaf Hachaim 66:24]
 2nd opinion in Admur ibid; 1st opinion in Michaber 66:4; Tur in name of Sefer Hamanhig
 See Kneses Hagedola 66; Kaf Hachaim 66:24
 The reason: As if one is allowed to stop in middle of Shema for the sake of the honor of a human and fear of him, certainly [one may stop out of honor for Hashem]. [Admur ibid]
 Admur ibid; M”A 66:8; Levush 66:4; M”B 66:26; Bach 66 [allows relying on lenient opinion]
Other opinions: Some Poskim rule like the former opinion, that even a Kohen is not top stop in the midst of Shema. [Michaber ibid “And the final ruling is like their opinion”; Rama does not argue; Beis Yosef 66; 135; Peri Chadash 66:4; Kesher Gudal 11:22; Kaf Hachaim 66:24 that so is ruling for Sefaradim] Other Poskim rule that one is to be lenient and make an interval during the blessings of Shema, which is only Rabbinical, but not during the reading of Shema which is Biblical. [Taz 66:5]
 Olas Tamid 66; Kesher Gudal 7:34; Beis Oveid 14:17; Siddur Yaavetz; Kaf Hachaim 51:25; 66:24
 M”B 66:24 in name of Elya Raba and Derech Hachaim
 M”B 66:26
 M”B 66:26; Ketzos Hashulchan 18:2; See Shaareiy Efraim 1:1; Kaf Hachaim 51:26
 M”A 135:6 regarding Birchas Shema; Beis Oveid 15; P”M 135 A”A6; Shaareiy Efraim 1:1; M”B 51:10 regarding Pesukei Dezimra; 66/27 regarding Shema and Birchas Shema; Ketzos Hashulchan 18:2 regarding Pesukei Dezimra; Kaf Hachaim 51:26; 66:24; Piskeiy Teshuvos 135:12
 Admur 51:4; 53:3 [regarding blessing on Tallis]; 54:3; Michaber and Rama 54:1; Tur in name of Seder Rav Amram Gaon; Kol Bo 4
 Hefesk Betefila 2:10 footnote 15 [p. 39] based on M”B 53:7 who prohibits initially stopping for a Mitzvah between Yishtabach and Yotzer; See Aruch Hashulchan 54:4; Kaf Hachaim 54:11; Igros Kodesh 8:99; Piskeiy Teshuvos 54:3 who all write or imply that one may not make an interval between Yishtabach and Yotzer Or even for the sake of a Mitzvah
 Michaber 135:5; Mahail in name of Rashba; 1st opinion in M”B 66:26; 135:17; Kaf Hachaim 66:24; This applies even according to the lenient Poskim brought in Michaber 66:4 and Admur 66:6 regarding if one was already called up, as nonetheless, they agree that initially one may not call him up. [M”A 135:6; M”B 135:17; Kaf Hachaim 135:28]
 Elya Raba 135:4; Beis Oveid 15; Tehila Ledavid 282:5; Kaf Hachaim 51:26; M”B 66:26; Ketzos Hashulchan 18:2
 M”A 135:6; Beis Yosef 135; Rashba; Maharik; M”B 66:26; 135:17; Kaf Hachaim 135:28
 The reason: As everyone knows that he is in the midst of Shema, and there is thus no reason to suspect for Pegam [that they will think he is not a Kohen]. Therefore, it remains prohibited to initially call him up, just as one may not stop and make an interval for other matters. [M”B ibid]
 M”A ibid due to Tircha Detzibura
 Rama ibid; Maharik Shoresh 9
 M”B 66:26; 135:17
 Opinion in M”B 66:26, in name of Yeshuos Yaakov and Nehar Shalom,
 The reason: As not everyone knows that he is in the middle of Shema, and this can cause people to think that his priesthood is invalid [i.e. Pegam]
 M”B 66:26; 125:17 as possibly even according to the Rashba one may stop Bein haperakim
 M”B 66:26
 See Ketzos Hashulchan 28:3; Hefsek Betefila p. 39 based on Admur 66:3 and 11 who is stringent regarding Bein Haperakim, unlike the M”B who rules one may say a blessing over a Tallis during Bein Haperakim. Likewise, Setimas Kol Haposkim imply that one should never initially stop to call up a Kohen or Levi who is past Yotzer Or.
 M”A 135:6 “Its possible that during Pesukei Dezimra one may stop”; P”M 135 A”A 6 that this refers to a case that he is the only Kohen in Shul; Beis Oveid 15; Shaareiy Efraim 1:1; M”B 51:10; 135:17; Kaf Hachaim 51:26; 135:28; Ketzos Hashulchan 18:2
 Admur ibid; M”A 66:8; Lechem Chamudos 2:21; Kneses Hagedola 66:5; M”B 66:26; Kaf Hachaim 66:24
 Admur ibid
The reason: In such a case, he is to rely on those opinions [Zohar 2 p. 206] who rule that he can recite the blessing over the Torah, over hearing the reading of the Chazan, as explained in Taz 141:3 [and Admur 185:2]. [Admur ibid]
 Shaareiy Efraim 1:1; M”B 51:10; Ketzos Hashulchan 18:2; Yechaveh Daas 4:11
 Beis Oved 17; Kaf Hachaim 51:27
 Admur ibid; M”A 66:8; M”B 66:26; Kaf Hachaim 66:24; See Elya Raba 66:7
 Ketzos Hashulchan 18:2
 M”B ibid
The reason: As one may respond due to honor of humanity. [M”B ibid]
 Admur ibid; M”A 66:8; Aruch Hashulchan 66:29; M”B 66:26; Kaf Hachaim 66:24
 See Aruch Hashulchan ibid that he is to resume Davening immediately after his Aliya, and not delay until after the next Aliyah, as is usually done
 The reason: As he willingly delayed the prayers, and it is thus not considered a forced interval at all [i.e. Oness], and without a forced interval at all [Oness] everyone agrees that he does not need to repeat from the beginning, as explained in 65:1. [Admur ibid]
 The reason: See Kneses Hagedola 66; Kesher Gudal 11:22; Kaf Hachaim 66:24; Piskeiy Teshuvos 135:12 that the entire dispute in Michaber 66:4 is in a case that the name was called, while if the name was not called, all agree that one may not go up. Now, according to the Ashkenazi custom to call up a person by his name and father’s name [See Piskeiy Teshuvos 139:2], he is not considered called up until his name and father’s name is mentioned. Hence, until that occurs it would need to follow the initial law that he may not be called up. Now, although once one was publicly motioned to come to the Torah, he must do so even if his name was not called, and if he refuses, the Talmudic warning befalls him [Brachos 55a, brought in M”A 53/22], nevertheless, in this case, since he is in the midst of Davening he is protected from the Talmudic warning [See Kaf Hachaim 66:24], and hence it once again reverts back to the initial ruling that he may not be called up. On the other hand, perhaps the mere fact that he was publicly motioned to go up, already enters him into the allowance to “answer for Kavod” even though he is not obligated to do so under the Talmudic warning, and he is thus to go up if the Gabai said his first or last name. Vetzaruch Iyun.
 As when one is privately asked, everyone agrees that he may refuse going up, and the Talmudic warning does not befall him. [See Chaim Sheol 1:13]
 Pashut, as this once again falls under the law of KJavod Hatorah.
 See M”B 66:26
 See Tehila Ledavid 282:3; Piskeiy Teshuvos 135:12; Hefsek Betefila p. 39
 Piskeiy Teshuvos 135:12
 Piskeiy Teshuvos 135:12 based on Tehila Ledavid 282:3 that only when there is Machlokes involved do we say the Maftor should also read; This is unlike Hefsek Betefila p. 39 who writes the Maftir may always read the Haftorah
 Tehila Ledavid 282:3
 M”B 66:27; Ketzos Hashulchan 19 footnote 7; Yabia Omer 7:9
 Based on M”B 66:26 who rules that one may give his name in middle of a Mi Shebeirach in honor of the congregation.