I've used snares, They are superbly efficient.
No bait needed & they work 24/7.
Just find the coyote hair on the barbs of a fence where they come under it on a travel corridor.
Hang the snare at head height and make the loop big enough for their head including the ears.
The traveling Coyote comes trotting through and Voila!
They choke out very quickly without the hours of agony felt by legholds, if you want something a tad more humane (?)
The bad news?
They work just as well on Rover, the neighbors dog.
If you have neighbors that let their dogs roam at night the snares will clean up that mess in short order.
Going out to check the snare with your kid and finding your own dog strangled might not go over so well either.
Best not to use them in urban areas as returning a dog with a sore foot beats returning one with rigor mortis and bulged out eyes.
They are also classified as "body gripping devices" and illegal in places that have banned trapping.
Again, I'd call your Animal damage control office.
They are there exactly for this type of thing.
They can legally use all of the above methods to remove the offenders.
Here in California since the do gooders banned the hunting of Mountain Lions, their numbers have skyrocketted.
A good friend is mine is our local animal damage control agent (government trapper).
He has killed 10 Mountain Lions around my house in about the last eight years.
The lions find 4H animals far easier to catch than deer.
The last one killed a neighbors goat and drug it under my deck a few years back while we were in Oregon deer hunting.
This one killed some sheep across the creek from my place...
(faces blacked out due to the fact that I did not ask them if they wanted to be on www)