A lot of the times, the trap is not dug up. Just dirt hole dug up and extra dirt put on trap, bait stole, or flat set moved around. Like fishing in leaves or grass and pulling the weeds off the hook every cast.
I just watched the Predator Control Group videos on how coyotes behave around a set. Very informative. The coyotes do not behave at a set like you think. In one instance, they had a coyote spend several hours at a set and never stepped on the pan. Often the coyotes got at the dirt hole from the sides or even behind the backing, digging and scraping everywhere but on the pan. Also, the coyotes almost always stayed on the high side of the set, even if that's where the backing was.
Might just be a coyote stealing your bait and making the mess after all. When I got into trapping I imagined the canines coming right up to the hole, sticking their face in it like a hungry dog and stepping right on the trap. Not so in the videos. I don't have much experience myself to say for sure, but if you think that's what is going on, you could try adding a couple traps to the sides or even behind the backing. And you might try rearranging the set so that the trap is where the coyote would be if he works the hole from the high side.
I also caught a coon in a Duke 550 that I had set for canines. This was before I saw the videos. I added a DP several feet away from the set in the direction where I thought the coon came from. I have since caught another coon and an opossum in the DP, but still nothing else in the original 550.