You have the right idea, skin them in the field.
I watched several utubes on skinning with the aid of air.
I rigged up an air bubble with an short hose and a basketball filler needle.
I inserted the needle in the leg, pinched around the back of the
needle and let er fill. It was good for about 2 coyotes.
That worked pretty well but the air bubble was bulky so I purchased one of
those portable tire pumps that work off the cigarett lighter. That works a lot better, so I installed a sockett in the back of my pickup bed in order for it to be reached with ease as everything is more difficult in the snow and cold and I like easy. Works great. No the hide doesn't just fall off.