I will gradually post more information on the service of Rosh haShana and other Jewish holidays.
After the reading of the Torah, and before the first set of blasts on the Shofar, we sing the poem 'Ét Sha'aré Raṣón ("Now, Gates of Divine Favor"), written by Rabbi Yehuda ben Samuel Abbas (c. 1145 - c. 1205). It describes Abraham and Isaac's struggle after God commends Abraham to sacrifice his Isaac.
Directly after 'Ét Sha'aré Raṣón, we sing "Adonai be-Qol Shofar" but not if this day falls on a Shabbat (as the Shofar is not blown on Shabbat).
The Torah scroll is returned to the Hekhaal (the Ark) while the congregation sings Psalm 29 to a tune that is only used for the first day of any double holiday: