Plus bits and pieces come out slowly, like Huan humiliating Sauron.
It's also worth thinking about how epic poetry is actually composed. Tolkien was obviously influenced by the Poetic Edda (which was anonymous), but there's other epic poetry: there's a theory that the Iliad and Odyssey existed in tellings already and Homer was the one who collected them, and the Mahabharata seems to have been collected and compiled over centuries.
Of course, Tolkien attributes the Lay of Leithien, in its whole complete form, to Daeron. We have no reason to doubt this: at least one Elf who definitively knew Daeron survives until the Third Age, when we know it has long since reached its final form (more elves knew him if we make the reasonable assumption that the Lay was established by the end of the Second Age). Elves also do not have the same time limitation as humans, and it is more reasonable to tell the whole Lay, from beginning to end, in a single sitting (which would have been totally impractical for something like the Iliad). But the Lay of Leithian does break up rather neatly into episodic bits once you get past the opening, which leaves open the possibility that Daeron composed the Lay out of order.
We also have to realize that, for a good portion of the story, exactly two people know the details of what happened: the two protagonists. So Daeron has to be getting these stories either straight from them, or through continued tellings of those stories. At some points, the only living people who know what happened are the villains (for instance, the story of the tricking of Gorlim is known only to Sauron and Thuringwethil, and perhaps Gorgol and Morgoth).
This was kinda rambly... I intended to present the idea that Daeron didn't compose the whole Lay, but it turns out that's a totally feasible thing to have happened... we just need to get him somewhere he can do it after this season.
