Do you have fast frames or similar? Love, love, love 'em for koozies. I use sticky stabilizer on them. Then when I embroider a koozie and am left with a "hole", I just patch it by laying tearaway on top of the hole (I use black but white would be fine too). Even if I didn't have the fast frames, I'd still use this method with a regular hoop because the sticky keeps the koozie in place and I can use the same sticky for 10 or more koozies. No need for spray, pins, etc.
No, I don't have the fast frames, but you should really try the Glad Press and Seal. It works very similar to the sticky stablizer, but it is better and a lot less expensive. I only hoop the press and seal once than I just patch the holes, much like you described. I LOVE IT. I have a couple sets of hoopless frames for my Esante in which I use the sticky stablizer, but for koozies I prefer the press and seal. I am sure a stablizer company will steal this idea and invent a similiar product once word leaks out.