There are three core parts to the first half of the Haggadah. The first relates to physical enslavement (genus) and redemption (shevach), followed by our spiritual enslavement and redemption. The second mentions how to fulfill the obligations of the night through Pesach, matzah and maror. The last is Hallel.
Why is Hallel broken into two sections? Praise and thanks are two distinct ideas. I can praise an outstanding individual for his qualities even though he never helped me. I thank someone who assisted me personally. We praise Hashem for creating the world. A blind person recites the blessing of Yotzer Or uvoreicChoshech, even though he personally does not benefit from the light, as it constitutes praise to Hashem. Sippur yetziyat Mitzrayim requires us to thank and praise Hashem for redeeming a small oppressed people from the grip of the most oppressive nation at the time. The first half of Hallel expresses thanks, hoda’ah, to Hashem and continues with the blessing of Asher Ge’alanu. The second half of Hallel comprises shevach. When we praise Hashem, we include all Hashem’s miracles in Nishmas. After all, the miracles of creation and sustaining creation are far greater than defeating the Egyptians. The last part of the Haggadah praises Hashem, and that praise is recited over the fourth cup of wine.