I have a number of guns that are pretty much just for hunting coyotes. Time of the year, terrain, frame of mind that day usually dictate what I'm shooting.
I love hunting with my little Sako Mannlicher in 222 Rem w/52gr Speer FB HP usually when the ground is bare and the shots are closer in
With snow on the ground and more open country my 22-204 w/hot stepping 40gr NBT(4000fps).
For off season, open country coyotes a 22-250 LVSF clone with again the 52gr Speer, the 6x45(55gr NBT0, 6mm-204(55gr NBT Lead Free) and the 25-204(85gr NBT) go along when I can't save the hides.
For thick country it is the combo guns 12ga/5.6x50R Mag(52gr Speer), 12ga/22 Highpower(55gr Buffalo Arms) and a 16ga/16ga/6.5x58R(90gr Speer HP resized to .260) The 12ga/22 Sav HP gets a lot of travel time as it has a spare set of 12ga/12ga barrels for impromptu bird hunts, SP and skeet.
They all do a great job of killing coyotes. TG weekend I took out a little Weatherby Vanguard Youth in 223(45gr HP White box Win) DRT'd the only coyote I shot at nice little rifle.
They all do the job asked of them and I have a built in gun locker in the truck so I usually have three along for the different types of terrain I'll be hunting or if I'm hunting with a partner.
This year is a slow year for me with the move to a new home in another state. My hunting vehicle is at my old place yet and I won't have it back until March so is most of my ammo hence the White Box 223. Look out next season, will have a new fur shed and I'm smack dab in the middle of coyote heaven.