Jeff,
As a taxidermist, the best advise I can give you is to get the animal in the freezer as fast as you can. Fox and coyote are especially prone to hair slippage, where coons are a bit more durable. Wipe off as much blood as you can, "tuck" the animal into a curled position to protect the nose, ears, and paws from freezer burn, then wrap it up in a garbage bag. Don't gut it, skin it, or wash it with water.
Sometimes you can do everything right and you can still lose the skin. I guy brought me a coyote this year with some mange on one leg. It later lost additional hair around the mange area and further up the leg and hindquarters. Sometimes fleas and mites in the ears will cause the hair to fall out too.
Good hunting!
