Aahh. A favorite topic!
In the 70's, when pelts were at a premium, I paid for a university education hunting coyotes. Being a budding gun nut back then, I finally settled on a Rem 788 in 22-250 with a Weaver 10X scope. I cannot remember if it was 52 or 53 grain Sierra's that would enter the coyote and usually not come out. Why i traded that rifle off, I will never know. I then moved on to a 6mm Model 77 that I never really liked.
With today's bullets (V-max, Ballistic Tip, etc), a light for caliber high velocity will work great. Especially if you dont care about pelt damage for the occasional through and through.
Back then, I tried a 222 and with any shot that was a little back (running coyotes seem to make for off placements), it didn't always do the trick.
Today, my first choice is an HB Rem 700 6MM with a 6.5-20x VX3 but that is just because I like the gun. Having shot coyotes with everything from 22 LR to 338 Win Mag, anything will work.
Part of the fun of being a gun nut is we each get to customize our rig to fit our exact game, caliber, and hunting areas (long range or calling in the critters).
Good luck with getting the perfect for you coyote gun.