In his commentary Zevach Pesach on the haggadah, the great Spanish scholar Don Isaac Abarbanel (1437–1508) answers this question.
But first, another question: Why is the Hallel is recited on occasions when we celebrate the salvation and redemption of the Jews? Why were these particular chapters of Psalms (113–118) chosen to express praise and our gratitude to G‑d? The Talmud explains that the Hallel mentions five topics of redemption: the exodus from Egypt, the crossing of the Red Sea, the giving of the Torah, the resurrection of the dead, and the birthpangs of the final redemption. The Talmud proceeds to explain where in the Hallel each of these topics is mentioned—explicitly or via allusion.
The exodus from Egypt, the crossing of the sea, and the giving of the Torah are all mentioned in the first two chapters of the Hallel. At the Seder, after reciting these two chapters, we say the Asher Ge’alanu (“who has redeemed us”) blessing, and we eat the matzah—both of which commemorate the miracles and redemptions of the past.
The following chapters of the Hallel mention the miracles which will happen in the future, with the coming of Moshiach—a topic of their own, worthy of being discussed separately. They are appropriately recited towards the end of the Seder, when we have just greeted Elijah the prophet, who will herald the coming of Moshiach, and when we focus on our anticipation of, yearning for and belief in the messianic redemption.
Have a happy and kosher Pesach!
Rabbi Baruch S. Davidson