Depends on if you need to kill them only, or if you are selling the hides.
Back in the 90s I bought a Toyota Land Cruiser for cash with money I earned on coyote and cat hides in only 5 months.
They were averaging $90 a hide for good ones.
When the value is up I would never use my 22-250. I much preferred my 222 Remington (SAKO)

Coyotes shot with my 222 brought me the full $90. Never lost one either, and I did make some very long shots. So don't believe that stuff about the 222 and 223 only being good to 250 or 300 yards. A 222 Remington is still supersonic at 640 yards. So at that range it's a bit better than a 22 LR at 10 yards. I think every kid who grew up with guns has killed a few coyotes with their 22s. I know I sure have and they all dies and they all died quickly. If you know your rifle and they round making long shots is not all that much harder with a 222 then irt was with my 22-250. I just had to hold higher.

Coyotes shot with my 22-250 brought about $40
Those shot with my $270 brought about $25

One rifle that was excellent for shooting hides is the 375H&H with 300 grain solids.
(No-----I am not kidding)
Those rounds are expensive so you can’t afford to miss many, but they don’t do much hide damage at all, and they kill the dogs quickly. A 375 Solid is a LOT better to kill coyotes with than a 22-250 if you are selling them.