My pharmacy would not just give you a new pack as a replacement, but I suppose every pharmacist does things a little different.
What will probably happen is this: as long as your wife still has refills, the pharmacy will just refill the script like normal. Then, the insurance will reject it as being "refill too soon"--they'll wonder why they should pay for it again when she should still have like 3 weeks of medication left. And as long as you've explained to the pharmacy staff why you need this early refill, the pharmacy staff will call the insurance company and ask for a "lost or mishandled medication" override. If your insurance allows for such an override, then you will pay your regular co-pay (it will not be free, you just won't be paying the cash price). If your insurance plan does not allow such an override, then you will have to pay the cash price. This is probably ~$25-$30 for a generic birth control, $50-$60 if it's something new and fancy like Yasmin or Yaz.
In the future, keep the OCPs away from the dog. Oh, and be sure to watch for blood clots. 🙂