What do I do when the accident happened between car insurance changes?

You have no coverage. Pay the bill. If you ask him to pay half, what is the legal basis for that? I doubt anyone would do that, so to avoid a judgment which will affect your credit for years, pay the bill.

If you didn't have insurance, you could lose your license if the other owner pushes things. You have little or no leverage in my opinion, and not much with which to negotiate. Other lawyers may see it differently, but other than offering 70-80% of the bill, I don't see a move here.

Interesting problem. If the guy was parked behind you so that you could not move, and if you had no way of locating him to make him move the car, you should have called the police and had the car towed. Not having insurance is a problem, because you are either forced into paying for the damage, or paying for a lawyer to defend you. I think that I would take a chance at defending myself IF he sues. Hopefully you have pictures to show that because the car was illegally parked you were blocked in. I think that under Michigan law, parking illegally may give you a defense.

Litigating a claim where you have no insurance means even if you are successful at establishing the other driver's fault, which sounds as if it is disputed, the most you can recover in damages is only your economic losses, i.e. medical bills and lost wages, because financial responsibility law prevents your ability to recover pain and suffering as a result of your not possessing valid auto insurance at time of accident. Therefore, would suggest small claims if you are going to pursue it at all.