So there are multiple good coaches available with zero interested franchises. Of course, they reject the minimum salary offer I give them, but if nobody will end up hiring them, won't I default hire from the best available coaches at the end of the coach hiring process?

Not necessarily. I haven't figured how the auto-hire system picks what coaches you get if you leave it all alone, but I wouldn't assume you get the best available or even close to it for minimum salary. (At one time, based on a number of hires I saw, I thought it chooses from coaches who weren't employed in the prior season, but I can't confirm that wasn't just an isolated trend.)

Wait till the final day of coach hiring. Keep an eye on remaining openings in your world, while you're at it. I've usually had luck sending out lowball offers - but not minimum salary ones - for the 11 AM cycle of the final day.

got it, thanks for the advice. I'll give that a shot