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The Mitzvah to blow Shofar begins from sunrise of each day of Rosh Hashanah until nightfall [צאת הכוכבים]. If one blew before this time but after the beginning of dawn he does not need to blow again. See Chapter 10 Halacha 6 for the full details of this subject!
Until what time may one blow Shofar? One may blow Shofar until Tzeis Hakochavim. [Regarding if a blessing is made when blowing past sunset-See Q&A]
Blowing within the first three hours of the day: If one is blowing Shofar on Rosh Hashanah without a Minyan in order to fulfill the Mitzvah, he is to beware to delay the blowing of the Shofar until three hours have passed from the beginning of the day. [This is commensurate to Sof Zeman Kerias Shema. Thus, one should not blow Shofar until after Sof Zerman Kerias Shema. Some Poskim write that in today’s times it is proper not to blow the Shofar even with a Minyan until after three hours have passed. All the above however applies in a case that one will be able to blow or hear Shofar later on past the 3rd hour of the day. If, however, one may not have an opportunity to hear it later on, then certainly it may and should be blown prior to 3rd hour of the day. Accordingly, those doing Mivtza Shofar to merit as many Jews as possible in the Mitzvah may begin doing so from as early as sunrise. Likewise, those who need to eat before Shacharis due to medical reasons may blow the Shofar even within Zeman Kerias Shema for this purpose.]
May one blow past sunset?
One may blow without a blessing from after sunset until nightfall.
Is a blessing recited? Some Poskim rule a blessing is not to be recited when blowing Shofar between sunset and Tzeis. Other Poskim rule that on the first day one may blow with a blessing between sunset and Tzeis. However, on the second day one must blow without a blessing.
 Admur ibid; Mishneh Megillah 20b; So also writes Admur 488:3 regarding Halel
 591:14; M”A 591:9; M”B 591:15; Kaf Hachaim 591:36
 The reason: The reason for this is because within the first three hours of the day G-D does not judge beyond the letter of the law, as during this time He learns His Torah which is called Truth [and thus judges people in accordance to Truth]. There is thus to worry that perhaps the supernal courts will investigate his actions and his prayers [Tekios] will be denied. However, after three hours into the day He judges the entire world and once He sees that the world cannot up stand to His judgment He therefore judges them in a manner of above the letter of the law, and the individuals prayer will not be differed. [ibid]
 Machazik Bracha 588:1; Mateh Efraim 588:2; Kaf Hachaim 588:6
 Admur 588:1; Chayeh Adam 141:1; Mishneh Berurah 588:1; Kaf Hachaim 588:2
 Chayeh Adam 141:1; 144:20; Mishneh Berurah 588:1; Kaf Hachaim 588:2; 600:12; Regarding Lulav: M”A 658; Regarding Megillah: M”B 687:5; Kaf Hachaim 687:9; P”M 692 M”Z 3
 Michtam Ledavid 17; Birkeiy Yosef 600:5, brought in Kaf Hachaim 600:11
Opinion of Admur: Admur ibid rules that from Alos until Tzeis is the permitted time for blowing Shofar. So, writes Admur also regarding Halel in 488:3. The source of Admur is from the Gemara Megila 20b which states that until Tzeis Hakochavim it is considered day. This seems to imply that during Bein Hashmashos one may blow even with a blessing! This is further inferred from the change of wording from the Tur which writes simply “until night” while Admur specifies Tzeis. Vetzaruch Iyun Gadol, as Bein Hashmashos is questionable whether it is night or day as rules Admur is 261:1, hence how can one say a blessing at that time.