No need to use stabilizer if you use a good quality felt.
I use felt for all my toilet paper covers and after much experimenting including wet and stick stabilizer,I prefer fusible stabilizer. i only hoop the stabilizer, lay the felt on top and press with the steam iron using a presser cloth. The iron fits in my 6x10 hoop and no hoop burns to deal with. Try it and let me know what you think.
I use stiffened felt to embroidery anything I want to remain 'rigid'. I find there is no need for stabilizer and it retains it's shape. Very easy to hoop and after you are finished you can steam out any bends caused by the hooping.
Locally I find stiffened felt at Michaels and Hobby Lobby with their regular felt products.