Let's say that a weird nationalist regime comes to power in Germany in the 1930s that is not Nazi and not against the Jews, but has a thing about Antarctica--they think it's Germany's destiny. They make a deal, we'll not launch a war, and we'll help the Brits and French restrain Japan, but in return we want all the uninhabited subantarctic islands and a huge chunk of Antarctica. And everyone says, why not?
If the Germans were able to pump resources into these islands, which no one else really wanted very much at that time, the resulting settlements could prove to be the eventual springboard to the continent as well as being valuable in themselves. Certain of those islands have enough high winds, consistently, to generate huge amounts of electric power; also the fast growing kelp in those waters, well, just read about what you can turn this premier renewable resource into (beginning with methane). Of course it would take the Germans several generations before they'd be able to mine seabeds in these heavy seas, much less mine under the glacial ice of Antarctica itself.
The northern tier of these islands does not have an especially harsh temperature, except for the wind chill factor; is ice free and mostly glacier free; has sustained sheep farming and even cattle at various times over the past century or more; has plenty of water; and one could probably develop a kind of high-wind small plot agriculture behind stone walls like traditionally existed in Cornwall. A few of the Aukland Islands of NZ even have forests that are remarkable adapted to the wind.
You're not talking about moving huge numbers of people there, but these islands, esp. Kerguelen, the size of Rhode Island pretty much, might be able to build up over time a population similar in size to Iceland.
To bring this about, you really need an Alien Space Bat who has a thing for the South Pole and uses a brain machine to persuade the Germans to channel their craziness away from Nazism and into a national enthusiasm for Antarctica, even persuading them that there's hole in it that leads to a Hollow Earth world. A silly idea, but at least I get to express my own enthusiasm for the subantarctic islands (not for the Hollow Earth).