If You don't want the Coyotes during the good cold Winter months when the fur is good simply find a local Trapper and give them to him,he may give You a little for them or at least use them so they don't go to waste!
Personally I like to use a lighter Caliber Rifle if I'm going to be Hunting Predators/Coyotes specifically.I have a CZ .204 using 39 grain Sierra Blitz King bullets and a new Savage Axis in a .223 using 55 grain Hornady V-max bullets!Both Calibers I shot are using lighter ballistic-tip bullets that usually have an entrance hole and no exit hole and are fur/hide friendly.
I usually like Hunting out of my existing ladder stands,ground blinds or an ocassional elevated shooting box!The colder the weather the more likely a Coyote,Bobcat or Fox will respond to a Call and as Nomercy mentioned late Dec and Jan is usually breeding time for the Coyotes so use the Female Vocals and such to draw a Coyote in for a shot!