Travis,
Make yourself a dog carrier! Carries just like a suit case and works very slick. I carry two of them in my pack. They will probably cost you almost nothing to make and take all of 5-10 minutes to complete. I will try to explain but it might be tough to get the picture of it to you.
Here goes:
Get yourself a 3/4" piece of wooden dowel or PVC. I used PVC and I think it is 3/4", just take the dowel or PVC, whichever you choose and hold it in your hand as if it were a handle for a suitcase. if if is comfortable in your hand the size is good. Cut it's length down so it is comfy in your hand, I would guess about 5-6 inches in length. Now drill hole all the way through the "handle" right in the middle. Make the hole just large enough for two pieces of rope to slide through the hole. Get a piece of rope about 20" long, tie it together with the ends and make a big knot. You have taken a single piece rope of 20" and now have a length of about 10". Make sense?? Now take that 10" section of rope and feed it through the hole in the "handle" all the way to the knot at the end. You may need a washer between the knot and handle to keep the knot from pulling all the way through. There you have it, one cheap but very handy Coyote Carrier. Just carry him like a suitcase and the only place he will rub on you is your leg.
Now to stringing that dead Coyote up with the new carrier. This may make no sense but is very easy to figure out once you have a Coyote on the ground.
I gather up all four feet but three will also work. Slip the loop of your carrier around both front feet about half way up the legs. Now grab one or both of the back feet and slip them through the loop also. Now, with 3-4 feet through the loop take the handle and start wrapping around the loop its self. Several wraps until you think you will have a good length to carry critter with. Not so short that it binds up on your hand but not so long that the critter drags on the ground while carrying. When you pick up the rope will slide up a bit and will cinch tight against the dew claws.
Give it a try, I think it works great. Once you get all cinched up you are set but getting the feet all together can be a pain. The only time it sometimes comes apart on me is when the tension is released as when you have to put him down to go over a fence or something. If some of this makes little or no sense give me a holler!
Good Luck,
Brent