My job search has gone more quickly than I expected and I will probably receive an attractive offer from a large, public US company in the next few days. I have a somewhat significant financial incentive from my previous employer to not start a new position for about another six weeks. The person who first contacted me about this position knows that, but I'm not sure everybody involved does. (It's not a secret.) The normal delay after acceptance in my location and industry is two weeks.
Ideally I'd like to either delay my start or receive some sort of offsetting compensation for the money I'd be leaving on the table. (I'd prefer the delay.) But I also don't want to start off on a bad foot with my new colleagues, and in the grand scheme of things, this incentive won't let me buy a luxury condo or a yacht or the like -- it's nice to have and nothing to sneeze at, but if I lose it, life will go on. I'd just be a little sadder.
How common is this sort of situation, and what is the best strategy for addressing it?