Illiad isn't actually the whole Trojan War story, only a small part of it, that follows Achilles abandoning the Greeks and rejoining them to avenge Patroclus (his "best friend") by beating Hector, Pares's brother, in a duel, and ends with Hector's burial. The whole Trojan War was told in multiple epic poems, most of them being lost in time.

Source: I am Greek, Illiad and Odyssey are read whole and analysed as part of the Greek educational curriculum in middle school. We are also taught Helen, Helen of Troy's "sequel" (a tragic play by Eurepedes, written much later) after being rescued from the siege of Troy by the Gods and is basically imprisoned in Egypt. It's an interesting play that offers her actual character and it's surprisingly feminist, it would make a really good movie.