pharmacy issues with birth control?

I have a prescription for the pill and it's written by my doctor for about 6-7 months at a time. However, my pharmacy will not let me refill it until I'm literally at the very end of all my active pills, and I'm trying to go pack to pack. They claim that it's an insurance issue, but why would that make any sense? For instance, I'll take my last active pill from this pack on Wednesday, but my "insurance" won't allow me to have a refill to pick up until Thursday, and that's if I'm lucky. My pharmacy said that my insurance won't cover a refill for another 7 days, but that they can try to get it for me in 5 days, so it's a maybe. Has anyone ever had this issue? If so, what can be done about it?