If your wanting a full body mount, it would be best to let your taxidermist do the skinning. If you want it tanned, you need to decide if you want a rug style or a cased style. To skin for either of them, start with the coyote hanging head down, make a cut down the back side of the hind legs, where the white of the under side meets with color, cut toward the hole just under the tail. No need to skin out to the paws, as they aren't used for cased or rug. So cut around the hind legs below the elbow joint. Once you have the legs skinned, skin around the rump so that you can get the tail cut off to keep ot attached to the upper part of the pelt. Once you have the legs skinned and the tail cut off, you can now start pulling the fur down, like pulling a sock off of the body. You shouldn't need to do much cutting with a knife until you get to the front legs. Case them out also, to below the elbow joint. Continue unrolling the hide down the the head. You will need to find the ears and cut below the hide so you end up with some cartlidge and then find the eyes and try to keep the eye lids/lashes attached, also keep the lips and nose attached.
It may take ya 30 minutes, or more, until ya do a bunch. I like to bag them and bomb the fleas before I skin, and it's easier to skin them while they are still warm!