I think there are a few tricks to skinning coyotes. One of the biggest for me is to be able to adjust the height. I hate working over my head or leaning over much. When I put up my first gambrel, I attached it to cable and through a pulley in the ceiling and down to a convenient place on the wall. I attached a 2 X 6 to the wall and ran lag screws into that. On the end of my cable I made loop. That way I could raise and lower the coyote at will. It helped keep the coyote at a comfortable height.
The other thing being able to adjust the coyote helps in is giving you leverage. I like to be slightly above where the hide and muscle meet. That way I can use my weight to push down. I'm to the point, I can skin one in just under ten minutes if I try to hussle. Most more likely at 12-15.
A couple years ago, I got adventurous and built a puller with a winch. It really doesn't save much time on a warm coyote but saves the back if you're doing more than a couple. On a cold or frozen then thawed coyote, it's a great help. Mine is tough enough that it will pull the hide off a semifrozen coyote. I've ripped a couple. Trick there seems to be taking it easy around any little cuts, especially towards the belly.