I put up most of our coyotes every year and this is what I do. After skinning it, I take it to the fleshing beam and scrape off the fat and big pieces of flesh. Don't forget to turn the ears (I do mine about half way up from the base). Then I'll sew up any holes that are in the pelt. Next I wash it good to get the blood and dirt out of it. Then I'll sling it around to get most of the water out of it and hang it in the shed to drip dry for a day or so (fur side out). Finally I put it on a stretcher skin side out for 2-3 days until it's partially dry. Take it off and turn it hair side out and put it back on the stretcher until it's dry. They'll keep this way for quite awhile until you get to the fur buyer.
If you want to keep one for display, you really need to have it tanned. Follow the same process except after you wash it, cover it with salt and let it pull the moisture out of the skin and then mail it to a tannery. If you have a taxidermist nearby who can tan it you can just take it to him/her after you've skinned it.
Mike
