Excess or Supplemental insurance has important differences from health insurance.
Basically, EXCESS pays YOU, and Health Insurance pays the Medical Provider.
Ergo, the CLAIMS process must be done quick and efficiently for Excess plans. You don't want medical service providers sending bills to collections. That can get ugly.
However, you can tell them about your Excess plan. This way, they know they'll get paid, and won't badger you as much. Further, you'll probably end up paying less.
(yes, Hospitals and Docs charge more to the UNinsured. Sucks, but that's the way it is...)
With regular health insurance, you'll get tons of paper from the company, called an EOB, or Explanation of Benefits. They'll tell you who they paid what.
With Excess, you'll need to gather the bills from the medical service providers, fill out a claim form, and MAIL it to the insurance company.
The faster you do that, the sooner you'll be sent a check to pay the bills.
The cool part is the coordination between the regular health insurance company and the Excess. Just include the EOB from your primary plan, and they'll calculate exactly what you need to clear all the debts.
This process requires diligence and patience. Get your agent involved. That's how he/she gets paid.