Let's go further with this. It's reasonable to assume that a UBI's cost would have to be offset by increased taxes, this would have to be progressive or the UBI would be pointless. So there would be a cutoff somewhere on the income distribution where the UBI payment would be offset by increased taxes, let's say at $80,000 to have a number. Making less than that, UBI would increase one's total take-home. Making more than that, the tax to support UBI would lessen one's total take-home.

The places with highly desirable real estate and high prices (e.g. SF and Manhattan) are already unaffordable to people making less than $80,000. So UBI would not increase the ability for residents to pay rent in those areas. Rather, it might actually decrease their net income, so by GP's argument it might lower rents in those areas. But in small towns (like City You've Never Heard Of, WV) with a low median income, UBI would increase net income. So it might increase rents there except those small towns are already emptying out as people leave to seek high-paying jobs in big cities, as you said.

So I think UBI would lead to rental/COL costs homogenizing somewhat rather than simply going up. It seems like this could solve several problems at once.