Most recently settled area habitable without modern technology

I'm going to start by pointing out that people with stone age technology have been successfully inhabiting a very inhospitable climate (the Arctic) for thousands of years before the Age of Exploration.

Archaeologists are certain that the predecessors of today's Inuit originated in the area of the Bering Strait, which separates Asia and North America. The first Inuit group, known as Paleo-Eskimos, crossed the Strait in 3000 BCE presumably on winter ice, which was long after earlier migrations by the ancestors to the North American Indians. Archaeological finds have revealed that the Paleo-Eskimos moved to the northern Canadian Arctic in 2300 BCE, apparently because of a change in climate. From there they gradually followed the herds of game across the Arctic to Greenland, and dispersed into more distinct nomadic tribes.

Given that, I'd say an "accommodating climate" is not a prerequisite for habitation, as the Inuit managed to inhabit what is arguably the 2nd least habitable climate on the planet (to say nothing of other Native American tribes and Arab nomads that inhabited "uninhabitable" deserts, or the Tibetans and Incas that established settlements at inhospitable elevations within the Himalayas or Andes). Wikipedia has a nice series of maps of pre-contact Inuit territory and settlement activities in the Canadian Arctic which leaves a number of rather large islands as unsettled.

Probably the most hospitable of these is Banks Island, which is home to enough wildlife to presumably support a small population of Arctic hunters, like the traditional Inuit. Its first permanent settlement was in 1929, when Sachs Harbour was founded. Since the Inuit managed to settle most of the rest of the Canadian Arctic Archipelago, they probably could have survived on Banks Island and the other nearby islands that have sizable wildlife populations.

Moving beyond the Canadian Arctic Archipelago, the Prince Edward Islands seem to be a survivable candidate as well. European seal hunters visited the islands because of their large seal populations, and a shipwrecked crew established a short-lived village in 1908 while awaiting rescue. A permanent research station was established in 1948.

Similarly, the Crozet Islands are another group of tundra islands that could probably sustain a small population of hunters in the Inuit style, being home to large populations of seabirds, penguins and seals. They're also notably temperate, as far as tundra islands are concerned. A permanent research station as set up on the islands in 1963.

The Kerguelen Islands are another interesting possibility. Along with the usual tundra wildlife (seals, penguins and other sea birds), the islands are home to an indigenous, edible cabbage species that is reportedly a good source of vitamin C, so there's the potential for some level of agriculture. The French made the first attempt at a settlement on the islands in 1877, with an attempt to set up a coal mining operation that quickly failed, and the next attempt at a permanent station on the islands was establishment of a research station in 1950.

Last, and probably least for the tundra islands, would be Macquarie Island. It has the usual wildlife for a tundra island, and served as a base for an Antarctic expedition between 1911 and 1914. A permanent research station was setup in 1948.

If you're dead-set on avoiding tundra islands, you don't have a lot of options, with the more habitable temperate or tropical islands being settled rather early.

The Bonin Islands are a collection of about 30 sub-tropical islands, 1000 kilometers south of Japan that were first settled in 1830. Their current population is around 2,500, so they're capable of supporting a reasonable population, many are forested, and have flora and fauna that would make them relatively easy to colonize with minimal technology, had technologically primitive people been able to get there, somehow.

The Tristan da Cunha islands are another possibility, being first settled in 1810, and having plenty of wildlife for hunting.

Another possibility is Île Amsterdam, but at 55 square km, it's probably too small. It does (or did) have the advantage of being forested, making it more habitable than most islands of its size or isolation, as well as having a healthy seal population, so it probably could have supported a low-technology population in the low hundreds. It was first settled (briefly) in 1871.

Ascension Island would be an outside possibility, being reported as largely barren at the time of discovery. It was used mostly as a stopping off point for ships to collect fresh meat from bird and turtles on the island, and has notable ground water springs to supply fresh water. It was first inhabited as a British garrison in 1815. Flora and fauna was eventually brought in, but presumably any stone age people who ended up there wouldn't be equipped to bring vegetation to the island.

Beyond that, there's not really anything that wasn't settled by 1800.