Fact: This reason for “half” Hallel is cited in later sources, but it is not the original one offered in the Talmud. The Talmudic reason relates to the Mussaf [additional] sacrifices offered in the Temple.
Background: The Talmud (Arachin 10a-b) lists those days on which Hallel is recited, and then asks why Passover is different from Sukkot: on the last six days of Passover only “half” Hallel is recited while for the entire Sukkot a full Hallel is recited. The answer given in the Talmud is that after the first day of Passover, the Mussaf sacrifices are identical each day, while on Sukkot a unique Mussaf sacrifice is offered each day. The Mussaf sacrifices are indicative of the character of the Yom Tov. By mandating the same sacrifice for each day of Pesach, the Torah is teaching us that the character of each day remains the same throughout the Yom Tov. However, the varied sacrifices of Sukkot reveal to us that each day of the holiday is different from the others. Therefore, in honor of the special meaning of each day of Sukkot, represented by each specific sacrifice, a complete Hallel is recited.
In an altogether unrelated discussion, the Talmud (Megillah 10b, Sanhedrin 39b) records that on the night the Egyptians drowned in the Red Sea (the seventh night of Passover), God prevented the angels from singing songs of praise since “His handiwork was drowning in the sea.” It would seem that this Talmudic account is unrelated to Jews reciting Hallel in subsequent years. The restriction on song was limited to the angels and applied only in that historic year. The Jewish people sang Az Yashir (Exodus 15:1-19) that night immediately upon emerging from the Red Sea  and in all subsequent years the angels sing on the seventh night of Passover.
Despite the clear reason offered in the Talmud for a “half ” Hallel on Passover, there is a source for this misconception. The Shibolei Haleket (13th century; cited in Beit Yosef Orach Chaim 490, starting verse kol ) quotes the otherwise unknown Midrash Harninu as suggesting that the reason for “half ” Hallel on Passover is because God prevented the angels from singing while the Egyptians drowned. The Taz (17th century; Orach Chaim 490:3) and Chavot Yair (17th century; 225) explain that because of this midrash, full Hallel cannot be said on the seventh day of Passover, and it would be inappropriate to say it on the intermediate days if it is not said on the seventh day. Many other commentators have challenged the validity of this recently added reason, and have questioned the need for a second reason in addition to the one offered in the Talmud.
The Torah Shleimah (Bo, 12:287) cites another, possibly related, reason for the reduced Hallel on the last days of Passover. The complete Hallel is only recited on a day called a “chag ” and the final days of Passover are never called chag; whereas all the days of Sukkot are so designated. The Yerushalmi (Sukkah, Chapter 5, Halachah 1, statement of Rav Yossi be Rav Bun) offers yet another reason for Hallel on Sukkot. The complete Hallel on the remaining days of Sukkot celebrates an additional aspect of the holiday, the arba minim (“four species” — lulav, etrog, hadasim and aravot) that are taken anew each day of Sukkot and that independently require the recitation of Hallel.
1. Hallel is the joyous recitation of Psalms 113-118. In “half Hallel ” the beginnings of Psalms 115 and 116 are omitted.
2. In the Talmud, God refers to the Egyptians as ma’asei yadai — the work of My Hands. In a contrasting phrase (in Sota 37a), as the Jews were about to drown in the Red Sea, God refers to them as yedidi — My beloved one.
3. According to Rabbi Elazar ben Rav Yossi (Pesachim 117a), upon emerging from the Red Sea, Moshe and the Jews not only sang Az Yashir, but Hallel as well. The Chavot Yair (225) even notes that, according to the midrash, the angels joined in singing Az Yashir with the Jews.
4. The statement that God prevented the angels from singing because the Egyptians were drowning is cited by the Kaf HaChaim (Orach Chaim 685:29) in the name of the Yafeh Lalev (3:3) as a proof that when there is destruction, even of evil beings, there is no blessing, and that is why there is no brachah on the mitzvah of Parshat Zachor. Rav Menashe Klein (Mishneh Halachot 7:81) has trouble with this assertion for, indeed, the Jews sang both Hallel and Az Yashir at the Red Sea, and only the angels were barred from singing. He therefore proposes several other reasons why there is no brachah on Parshat Zachor.
5. For a full discussion of this, see the weekly parshah sheets Torah Lodaat by Rabbi Matis Blum on Tzav and Pesach, 1994. See Torah Temimah to Exodus 14:20, note 9, who observes that merely omitting two half paragraphs from the Hallel leaves a significant song of praise in place that would not meet the objective of curtailing joy, as seemingly required by the non-Talmudic reason for “half ” Hallel. See, however, Rav Aharon Kotler, Mishnat Rebbi Aharon (1996; vol. 3, page 3) who offers an alternative understanding of this second reason as applicable only for the seventh day of Passover, while the Talmudic reason relates to the days of Chol Hamoed (the intermediate days) of Pesach: therefore both reasons are needed, and neither is superfluous. It seems the Taz disagrees and holds the midrashic reason sufficient. cf. Mishnah Berurah 490:7.
6. Accordingly, the Mishnat Yaavetz (by Rabbi Betzalel Zolti, Orach Chaim 20:4) rules that if one did not have a lulav in the morning when he recited Hallel, he should repeat Hallel later in the day when he performs the mitzvah of lulav. He argues that the mitzvah of lulav warrants its own independent Hallel, even if Hallel was already recited in honor of the festival.
Reprinted from JEWISH ACTION Magazine, Spring 5760/2000 issue
This material is for study purposes only and should not be relied upon for practical halachah. One should consult his own competent halachic authority for specific questions.