I will try to describe how they set foothold traps and foot snares in the snow in Sweden and Russia. I watched it done with a foot snare while in Sweden one January, and a Russian friend of mine showed me how they do it in Siberia with footholds.
I have never seen this method used or described here in North America.
Both my Russian and Swedish friend told me that when an animal such as a fox or presumably a coyote goes through the snow they will step in the same hole in the snow (track) that they stepped in the last time they came down the same trail.
The trapper selects an area along the animal's trail where he wants to set the trap. He then approaches the animal's trail from a right angle preferably from behind or alongside a tree or bush. Starting maybe 16 inches away from a particular footprint the trapper carefully scoops out a hole in the snow maybe a foot square with a wooden paddle type of tool made for the purpose; the snow is set aside for filling the hole latter.
Once the hole is dug, then the trapper uses the paddle to slice under the snow a few inch towards the footprint until he come right under the footprint at the level where the animals foot cam to a rest. The paddle is then wiggled side to side a bit in order to excavate a place for the trap to set, the trap is slid into position so that the pan is directly where the animals foot will come down the next time it steps in it's old track.
The trap chain is buried back towards where the trapper is standing along with the drag. The paddle is then used to fill the hole back in that was originally made, and as the trapper backs away he uses a large feather duster to smooth everything off as well as uses the paddle and duster to fill in his boot tracks. No scent or anything else is used that may cause the animal to alter it's original course.
I wish I had a video of this type of set. It's very simple and makes a lot of sense when you stop to think about it. A little new snow or wind makes it even better.
I just found two old scanned pictures I took of the final steps of the process. The image on the top a stick is pointed at the track under which the trap pan lays. The picture on the bottom is dusting the area over upon retreat. Notice in the picture the trapper is standing on the edge of a road where the animal crosses, an easy place to disguise your presence from the animal.