I'm strongly considering this, or at least building a buffer for this ahead of actually stepping away from work. I'm currently in MCOL for family reasons but don't plan to stay for life. I'd guess you're most likely to see something like that or people moving before FIREing compared to doing something like this after the fact.
I'd hope that if I did under prepare I would just take a part-time or low stress gig to cover the difference in rent/payments. Unless you're moving from LCOL or MCOL, if you're willing to rent it's probably unlikely you're talking about an excess of + $3000 a month at the absolute highest end. That's especially true if you're willing to downgrade in terms of size, quality, or location. Maybe if you're going from VLCOL to Manhattan or downtown SF or something.
My current budget: $2300 a month, of which $800 is rent for a 1br.
I'm much more likely to do something like, at the 4% rule:
min budget to consider safely FIREing: $3333 a month
FIRE and HCOL is in play: $4000 a month
At $4000, that'd give me $1700 a month in lifestyle inflation room. You're not going to be in a penthouse but you can get a 1br almost anywhere for $800 + $1700 = $2500 a month. Other things cost more too of course, but that's not really true for the things I spend my money on (i.e. not restaurants and bars). It's also not hard to imagine instead spending just $2000 a month for 3 years, resulting in $18k of extra fluff (any gains), which would support staying in a $3000 a month on housing for the following 3 years or a single year in a $4000 place etc. Even if I wanted to stay in a $4000 place I only need to make up $1500 a month or $18k a year...