Health insurance claim filing question

I saw a specialist as a follow-up to a mammogram. I asked for the follow-up (the mammogram turned up nothing, but I wanted advice on the problem I was still having). The follow-up visit happened after my new insurance became active, but before my old insurance was canceled, so at the time of my appointment, I was covered by 2 insurances.

The visit claim was filed with my old insurance (which is a crappy insurance) and I got a bill for charges that weren’t fully covered.

Can I try filing it with my new insurance to see if the full amount will be covered? How would that work if it was possible? Would the old insurance get reimbursed for the money they already paid? How would that happen?