When I go hunting coyotes on purpose, I do use more gun. Either a .223, .280 Rem, or tight choked 12Ga with buffered lead BB.
They do, however, often present themselves as targets of opportunity around the farm. Since I can't/won't keep a centerfire rifle in the barn, an old, inexpensive .22 is pressed into service. I have folded them cleanly to 125 yards with CCI Stinger, misc. 38 gr HV hollow point, and Browning high performance 40 Gr HP. The latter is my favorite. Better accuracy, impact, trajectory and wind than the Stinger or generic 38 HPs. I believe Winchester has an identical load. Every one I have shot with this load or Stingers died quite quickly, just minus the shock and awe of the high velocity centerfires. Of the few I took with generic 38 HPs during the great rimfire shortage, results were mixed. Didn't lose any, but one required 3 hits at relatively close range 50-75 yards.
When fur prices were decent, I did a fair amount of trapping and coyotes and bobcats were on my list. My standard trap finishing load was Aguila 38 gr Subsonic HP from a Ruger MK2. This load was 100% lethal, first shot, in the head at close range. More substantial rimfire cartridges are IMHO sufficient for coyotes if your marksmanship and self control on shot selection are up to the task.