I'm in the middle of getting approval for my hysto and found out that in my city, which is a really big city, my insurance has exactly one contracted physician to do the dirty deed. My case manager is trying to find out if I can get my preferred doctor covered instead. The answer might ultimately be "no," but I'll have to wait and see.
Since I have already met my deductible, I will pay a $1500 copay even if I go with their surgeon, but I will obviously pay more if I go with my own surgeon and they decide he isn't fully covered. However, the amount I pay depends to a certain extent on what I've already paid out on other services this year. The copay for this surgeon is normally $3000, but that amount is offset by my payments all this year. So he might run $2500 or even less.
It really depends on the specifics of the plan, so you are best off if you gather all of your information and questions and then call your insurance company.