The order when two Torah scrolls are removed:[1]

It is forbidden to roll a Sefer Torah in front of the congregation due to respect for the congregation.[2] Accordingly, whenever two different sections of the Torah need to be read, such as by every Yom Tov[3], Shabbos Rosh Chodesh[4], Shabbos Chanukah[5], the four Parshiyos[6], then two scrolls are to be removed.

What is a congregation to do if only one Sefer Torah is available?[7] If two scrolls are not available, then one is to roll the Torah scroll to the second reading after the first reading is complete.[8]

The following is the order:[9]

1. The Torah scroll is to be rolled to its proper section before Davening in order to prevent needing to roll it in front of the congregation.[10] If this was not done before Davening, then it is to be done during Davening, and may be done even during Kerias Hatorah.[11]
2. Both Sifrei Torah are to be removed simultaneously from the Ark.[12]
3. Holding onto the second Sefer Torah:[13] While the reading from the first Sefer Torah takes place, the second Sefer Torah is held. [It is to be held by the person who performed Pesicha[14], or given to someone else to hold.[15]]
4. What to read in the first scroll? Whenever two scrolls are removed on Shabbos or Yom Tov, and the second scroll serves for the reading of Maftir, the weekly Torah portion is read from the first scroll until Shevi’i [on Shabbos and until Chamishi on Yom Tov].
5. Saying the Kaddish: On Shabbos and Yom Tov, after Shevi’i/Chamishi, the second scroll is then placed on the Bima[16] [to the left of the first Sefer Torah, near the Baal Korei[17]] which is followed by half Kaddish.[18] [If the half Kaddish was said prior to placing the second Sefer on the Bima, it is nevertheless valid.[19]]
6. Hagbah is then done to the first scroll.
7. The Mi Shebeirach for the ill is customarily recited after Hagbah.
8. One may not open the second scroll until the first scroll is rolled up and placed in its Meil.[20]
9. The portion of Maftir is read from the second scroll. The half Kaddish is not recited after this Aliyah, having been said earlier.[21]
10. One then performs Hagbah to the second scroll and reads the Haftorah.
11. On Shabbos Mevarchim, the Chazan for Musaf takes the scroll of Maftir to hold.[22]
12. When the scrolls are being returned to the Aron, the scroll of Maftir is taken first.[23]

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[1] Admur 488:6; Michaber 144:3; 488:3; Tosafus Megillah 30b; Rosh Megillah 2:10

[2] Michaber 144:3; Admur ibid; Yuma 69a

[3] One must read from the Parsha of the Holiday, as per the institution of Moshe Rabbeinu, and one must also read the Karbanos from Parshas Pinchas, as per the institution of the Rabanan Savuraiy.

[4] Michaber 425:1

[5] Michaber 684:2

[6] Michaber 685:2

[7] Michaber 144:3; Ritva Yuma ibid; Hagahos Mordechai Gittin Remez 463; M”A 144:7; Mamar Mordechai 144:5; Elya Raba 144:3; Kaf Hachaim 144:17

Other opinions: Some Poskim rule that in such a case one may not roll the Sefer Torah and read the second portion from it, and is rather to read the second portion from a Chumash. [Rashba, brought in M”A ibid and Beis Yosef 144; Peri Chadash 144:3]

[8] The reason: As in such a case we differ Kevod Hatzibur which prohibits the rolling of the Torah scroll in their presence. [Michaber ibid] As it is better to differ Kevod Hatzibur than to differ the institution of our Sages. [Beis Yosef 144; Kaf Hachaim 144:19] Alternatively, it is allowed as we assume that the congregation forgives their honor so they can hear both readings. [M”A 144:7]

[9] See Mishneh Berurah 685:9 and 12; Piskeiy Teshuvos 147:10-11

[10] Pashut as this is the entire reason for why we are to take out two Sifrei Torahs.

[11] See Q&A!

[12] Rama 147:8; Or Zarua in name of Yerushalmi

[13] Rama 147:8; Or Zarua in name of Yerushalmi

[14] Siddur Yaavetz that the person who took it out should hold it and not give the Mitzvah to another unless he is too weak

[15] See M”B 147:29

[16] Rama 147:8 that one is not to remove the second Sefer Torah until the first one is placed down; M”A 147:12 and M”B 147:27 that it is to be placed before Kaddish

[17] Mishneh Halachos 13:22; Piskeiy Teshuvos 147:10; Some however say that the Torah is to be placed on the right side of the first Torah. [Rav M. Harlig; See Hiskashrus 1078 footnote 5]

[18] Admur 282:14; Rama 282:4; M”B 147:27

[19] Piskeiy Teshuvos 147:10

[20] Michaber 147:8; Kol Bo; Elya Raba 147:9; Kaf Hachaim 147:42; Ketzos Hashulchan 84:2

[21] This follows the Ashkenazi custom. However, the Sephardi custom is to say two Kaddeishim whenever two or more Sifrei Torah are removed. On a day of two Sifrei Torah, the first kaddish is recited after the first scroll, and the second is recited after the second scroll. [Kaf Hachaim 684:19; Yalkut Yosef 144:8; Piskeiy Teshuvos 147:9; 282:7] See above Halacha 1B!

[22] Shaar Efraim 10:41; Piskeiy Teshuvah 150:4

Bedieved: If the Chazan accidently took the first scroll, it is not to be switched for the second scroll due to worry of Pegam. [Shaareiy Efraim ibid]

[23] See Shaar Efraim ibid

[24] M”A 144:6; Rav Yaakov Libeis Levi 82; Kneses Hagedola 144; Kaf Hachaim 144:12

[25] See Piskeiy Teshuvos 134:4

[26] Sefer Revid Hazahav 2:15

[27] Kinyan Torah 4:16; Piskeiy Teshuvos ibid; See Maharsham 6:3; Yabia Omer 4:15