Popapi,
I don't know for sure, but I venture to guess that the most widely used round to take coyotes would be the .22-250. I've got the Rem. 700VSSF with the muzzle crowned Savage style.
I fell in love with the VSSF's back when I attended a Sniper/Observer course for my dept. back in the late 90's. They bought me the same gun in .308 and it shot unbelieveably.
The Savage's, now with the advent of the Accutrigger, are superb shooter's for the money.
As far as a quieter round to hunt coyotes with, I've got the .223 and others, but I just had a custom barrel chambered in .22 Super Jet for my Encore. It's a non-commercial wildcat and the cases have to be made from .357 Magnum brass.
The ballistic's are slightly better than the .222 with the same weight bullets, and on approx. half the powder charge to boot.
The velocities are only slightly lower than the .223. I use about 14.3grs. of Li'l Gun and the 40gr. VMax, at 3400fps.
Just a fun gun......!
Take care,
Bob