You could ask this question about several different types of "precursor", so I'll try to cover a range of bases.

• Folktale sources and cross-cultural cognates. there are several episodes that have cognates in other cultures and other mythologies. For example, the Cyclops episode in Odyssey 9 corresponds closely to two folktales, "The Ogre Blinded" (AT 1137) and "Noman" (Thompson K602), and puts these two stories together in a unique combination. A condensed summary of folktales that have exemplars in Homer can be found in William Hansen's chapter "Homer and the folktale" in Morris & Powell (eds.) A New Companion to Homer (1997), pp. 442-462; for more details his book Aradne's Thread (2002) is a good reference source. The vast majority of folktale parallels that Hansen identifies are in the Odyssey; only two are in the Iliad: the stories of Meleager in Il. 9 ("Meleager"), and Bellerophon in Il. 6 ("Potiphar's wife"). More literary kinds of cross-cultural parallels also exist: the bow contest in Odyssey 21 has parallels in both the Ramayana and the Mahabharata (more details in this post from a couple of weeks ago).

• Independent testimony to the myths. Greek and Etruscan vases with legendary themes going back to the 8th century BCE give us good evidence of a set of traditions about the Trojan War legend that are either independent of Homer, or at most only loosely connected to Homer. It was only in the late 6th century BCE that the Homeric epics acquired a level of prestige that made them the definitive canon for the stories they report; prior to that, when we look at pictorial depictions of Trojan War scenes -- and there are a lot of them -- they are effectively independent testimony about the myths. There is some wiggle room, however, and some scholars would not want to remove Homer completely from the influences feeding into vase paintings. But as there's no evidence at all prior to the mid-500s that anyone was influenced by Homer specifically, as opposed to being influenced by non-epic myths, or by the language and forms of the epic tradition, their position still assigns more importance to texts than the evidence requires. More details and references in this older thread. A good example of a storyline that we know primarily from pictorial art would be the story of Troilos: Achilleus' pursuit and killing of Troilos is a very popular theme in early Greek art, but scarcely even mentioned in any textual source until the Byzantine era.

• Literary precursors. A lot of work has been done in the last 50 years on Homeric adoption of motifs from pre-existing poems or poetic traditions; this sub-field is called "neoanalysis" (for reasons that aren't important right now). These "motifs" may be either isolated tropes, set-piece scenes, or entire plot-lines. Our evidence for the stories that appeared in the source material is all later than Homer, which complicates things, but there are certainly cases where the most parsimonious explanation for parallels between Homer and other poetic texts is that Homer was borrowing from an earlier form of those texts. The main corpuses of source material were, for the Iliad, the stories surrounding Achilleus' death (and our earliest evidence for those stories is the fragments of Arktinos' lost epic the Aithiopis, which probably dates to a handful of decades after Homer); and, for the Odyssey, the Argonauts and the adventures they had on their voyage to Kolchis (but our evidence for the Argonauts story is much later).

In some cases these parallels are very close. For example, we know that in Arktinos' Aithiopis there was an episode where Nestor's chariot was disabled by Paris shooting one of his horses; Nestor was in immediate peril from the Aithiopian hero Memnon, who was aiding the Trojans, but he got rescued by his son Antilochos. In Iliad 8.80-115, Nestor's chariot is disabled by Paris shooting one of his horses, and he is in immediate peril from Hektor, but he gets rescued by Diomedes. The Aithiopis doesn't survive. But Pindar's Pythian 6 gives us an account of the Nestor-Memnon-Antilochos episode (tr. Verity):

In former times, too, this thought was held by mighty Antilochus,
who stood firm against murderous Memnon, the Ethiopians’ captain,
and died to save his father; for Nestor’s chariot was encumbered
when his horse was wounded by Paris’ arrows.
He was shaking his mighty spear,
and the old Messenian’s mind was confused;
he called out to his son, nor were his words flung idly to the ground.
That godlike man stood firm just where he was...

The scholia (ancient commentaries) on this passage actually mix the two episodes up, and quote from Iliad 8 to explain the Pindar passage. But thanks to other sources about the Aithiopis, especially Proklos' summary of that poem, we know that Pindar is referring to events in the Aithiopis.

The question then becomes one of priority: which came first, the Homeric episode or the parallel episode (in this case, the Aithiopis)? That question cannot always be answered definitely. In this case, priority is usually assigned to the Aithiopis episode, for a number of supposed reasons that are not actually very compelling. I agree, though, that that Aithiopis scene probably does have priority: not because of the difficulties that some neoanalysts believe exist in the Iliad passage (which supposedly imply that it is shoehorned into the Iliad), but because the point of the scene is to provide the motivation for the saviour's death (Antilochos gets killed by Memnon); that plot element is what makes the episode an organic part of its context, but that's what is missing in Homer. That's not totally compelling, of course. But in a handful of cases there are very good reasons to assign priority to the non-Homeric version: there are also close parallels between the deaths of Patroklos and Hektor in the Iliad, and the death of Memnon in the Aithiopis. In the Aithiopis the motivation for the storyline is to lead up to the death of Achilleus; the Iliad points to the same motivation (the poet harps on about Achilleus' death repeatedly), but it never actually takes place. So that case is much more clear-cut: the Aithiopis version definitely has priority in that case.

But a few points need to be stressed: (1) It used to be thought that neoanalysis proved that Homer was later than the Aithiopis. No one believes that now (or no one sane). Instead, we're looking at borrowings from a proto-Aithiopis, not from Arktinos' poem (and similarly for other poetic sources). (2) We're not talking about Homer taking someone else's poem, editing the beginning and end of it, and slotting it into his own; we're talking about adaptation in a literarily creative way. (The "slotting a poem in" model probably does apply, however, to Iliad 10, which almost certainly was an independent composition. But that's the only case.) (3) None of this has any bearing on the relative literary merits of any of these poems. A poem that plays off other poems is not inferior for that reason; quite the opposite. Equally, we know basically nothing about the literary merits of the Aithiopis.

This answer is probably a bit more abstruse than you were hoping for. Bear in mind that the evidence we have for 7th century BCE Greek literature is very incomplete and very much shrouded in mysteries and doubts: trying to make deductions from the scattered hints we have is necessarily abstruse. If evidence on this question were easy to find, you probably wouldn't have been nearly as interested in asking it!