Rosh Hashanah 4:8

Mishnah

The one that (prays) in front of the teivah on the Yom Tov of Rosh Hashanah: the second one (by Musaf) blows (the shofar), and at the time of Hallel, the first one (by Shacharit) recites the Hallel.

Yerushalmi

The Shofar is during Musaf because of an incident that occurred that the enemies of the Jews heard the shofar and thought that it was a war cry, and they came and killed the Jews, but since they see that the Jews are reading Shma, davening, and reading the Torah, they figure that the shofar is another religious ritual.

If so, why not delay Hallel as well until after Musaf?  Because not everyone is in the synagogue.   But the same can be said regarding the Shofar.  (So the proof of everyone being at the shul is rejected).  The Yerushalmi gives a drash on a pasuk in Yeshayahu 58, and Tehillim 17.  The conclusion given is that Rosh Hashanah is different because the mitzvah of the day is by Musaf, and therefore Shofar should be by Musaf.  There are then 2 drashot given, one about a diyuk by the korbanot of Rosh Hashanah and then one by Atzeret.

Bavli (32b)

Hallel is early because we should do all mitzvot as early as possible.  Why is Shofar different? Because there was a decree from the kingdom (perhaps hinting at the ma'aseh from the Yerushalmi above.

(Mishnah) and at the time of Hallel, the first one (by Shacharit) recites the Hallel.

Here we pay attention to the phraseology of the Mishnah: "at the time of Hallel".  That means that Rosh Hashanah doesn't have Hallel.  Why?

Rebbi Abahu says: The ministering angels said in front of HaKadosh Baruch Hu: Master of the World!  Why aren't Israel singing songs of praise before you on Rosh Hashanah and Yom Kippur?  He said to them: Is it possible that a king would sit on the chair of judgment (with the) books of the living and the books of the dead open before him, and Israel are singing?