This is an extremely difficult question to tackle - but interest in this subject is such that it attracts a range of efforts.

First, we are not likely to be able to link legends (stories told generally to be believed) with actual events tied to the last glaciation. The best case I have seen are efforts to see references to glacial-era islands that disappeared with rising ocean levels: such stories have been recorded in Australia and may be linked to events that occurred roughly 12,000 years ago. To achieve this sort of continuity in oral tradition, one would need an extremely stable population and culture. What exists in indigenous Australia is largely unmatched elsewhere, so finding traditions that speak specifically to one's own backyard as legendary accounts, reaching back to the Pleistocene, is not likely to be very common.

That said, there have been efforts to track folktales (stories that are typically told as fiction, not to be believed) into a deep past. Some of these efforts follow the so-called phylogenetic modeling of scholars such as Julien d’Huy. He has used a method comparable to what is employed by geneticists to track variations in widespread narratives to determine age and origin/distribution. When he can claim to have evidence of narratives from both hemispheres, he can advance his hypothesis that a story descends from a common ancestor that predates migrations to the New World - pushing its origin back into the glacial period. He uses this approach with the Polyphemus story (Tale Type ATU 1137), which appears in the Odyssey when Ulysses escapes from the cyclopes. Not everyone agrees with this method, but it offers some of the most convincing evidence of something that may have survived that long.

Using a different method, there have been efforts to suggest that there is a similar distribution for a story about a great hunt, a story associated with Ursa Major (the Big Dipper). According to these range of stories, there was once an epic hunt that was so remarkable, it is memorialized as a constellation - sometimes the Big Dipper and sometimes the Belt of Orion. This analysis is also thought provoking but not accepted universally.

Opportunities to pursue stories along the lines that your question seeks are extremely rare: most stories have more limited distribution; even when similar stories are found on Eurasia and the Americas, we must question as to whether we are looking at independent invention rather than diffusion and descent from a common ancestor. Nevertheless, the stories pursing this question are not without their merit and I find them quite exciting.