The biggest challenge is that the Arctic is an absolutely bountiful land compared to even the Antarctic Peninsula, reason being because of the ample plant life and nearby forests which increase the amount of usable tools and the biomass. Antarctica itself has none of that, and all of the islands south of the Falklands (and definitely those south of South Georgia) are rather impoverished in flora and fauna compared to the mainland. Compare to Ellesmere Island which has numerous edible plants and populations of fox, bear, rabbit, and reindeer. As they were on the mainland, it seems fair to compare the Yaghan, the southernmost native group, to the Inuit, and not to the more populous and socially complex Inuit in Alaska either. This difference can probably be attributed to the fact that the Yaghan (and their cultural relatives to the northwest, the Kawesqar and the Chono) had to develop their culture more or less independently while Inuit culture can be traced back to thousands of years of development and refinement among many different peoples in North Asia.
But let's imagine that for whatever reason things are different. Maybe the southern Chilean coast evolves farming communities (potatoes were domesticated near Chiloe, where the Chono people traded with) which spread to southernmost Tierra del Fuego and produces an alt-Yaghan who mix farming with fishing. They'd have more room to culturally experiment and perhaps develop things like whaling. We can imagine these alt-Yaghan would spread to the Falklands and South Georgia, transplating their lifestyle and probably terraforming the island with trees from the southern forests, probably innovating complex windbreaks to do what farming they could amidst the windswept tundra. By they time these alt-Yaghan spread to the South Sandwich Islands, they'll find growing trees is practically impossible and farming simply not worth it. Perhaps some beneficial plants could be transplanted. But with skin boats and what wood they can find, they'll be able to continue whaling and sealing. Likely beneficial genetic changes would occur in these populations--similar to those that occurred in the Inuit--optimising their body for a diet heavy in the meat of whale and seal and low in plants.
South of that is a lot of open sea, but perhaps the alt-Yaghan could still discover South Orkney and the South Shetland Islands. I'm not sure what sort of useful native plants of Tierra del Fuego could survive there--probably not many, and definitely no trees. There also appears to be a dire lack of driftwood, which for the Inuit was critical for making hunting tools and the frames of their boats. I'm not sure if bones from seals or whales could be a reliable substitute. Given their isolation, lack of reliable trade (due to ice and storms), and incredibly meager resources, it's very possible this is a step too far for any hunter-gather culture to adapt to.
If we assume a culture of sealers/whalers could survive on those islands, then they could probably colonise the Antarctic Peninsula at a very low population density, camping in sheltered coves and surviving entirely off meager plant resources they transplanted, algae, and whatever they can kill. Like some cultures in the Arctic (including IIRC the earliest inhabitants of Greenland and even those of Ellesmere around 800 years ago), they'd be prone to sudden extinction caused by climate change. If they survived the Little Ice Age, they'd probably be wiped out within a generation or two by sealers, which would be a tragic end to what would hands down be the toughest, most resilient culture out of any that has ever existed in human history.
I'd wager the most likely place to support any population would be the Antarctic Peninsula, being one of the 'warmer' (as in it sometimes gets a couple degrees above freezing in the summer) areas of the continent. It's also relatively close to South America, though it's still quite a distance across the Drake Passage. Did any of the indigenous peoples living in the area have the ability to cross that area, either accidentally or deliberately?
While the Yaghan and their cultural relatives to the northwest had a maritime lifestyle, I don't believe their canoes were sturdy enough to survive the trip. And if they were unfortunate enough to wash up on Antarctica, they'd be dead within weeks because of how harsh the land is.
