I had a loose one also. It looked as though the plastic mirror bezel has legs on it that protrude through holes in the sun visor center material. the plastic legs looked to be melted at the ends so they could not pull back through. My car had several that had pulled through due to the ends of the legs breaking off. I drilled a small hole through the bezel and inserted a small screw through the hole and screwed into the center material. That kept the mirror in place until I replaced the bad sun visors. My wife found a deal on a new pair for $100.:thumbsup:
You can find the used stock ones cheap at places like Corvette Recyclers or Parts Taxi (and of course on eBay), but all the sweet ones that I have seen were at Bloomington Gold and Corvettes at Carlisle. They had leather covered ones with vanity mirrors that were cheaper than the new, stock ones.