First key question... What is legal in your area?
Many areas are Rimfire only for small game and/or varmints....
If this is the case - look at a 22 mag or a 17hmr.
2nd key question.... What sort of ranges are you talking about? If you are talking coyotes in thick brush or under 50 yards and it's legal... Shotgun and load it with shot between #6 and #3. They move QUICK and a shotgun gives you a bit more wiggle room for error.... They usually don't stand around long enough to give you time to settle in and get the scope on target... They pop in and disappear...
Now.. If you are talking coyotes out at longer ranges - and you will be hunting from a stand or fixed position - and hunting them with Center Fire is legal... 223, 204 ruger, 243, 7.62xanything, 308 even... ... but I certainly wouldn't turn my nose down at a 17 HMR in a high quality rifle - it's easily capable of 300 yard kills and they tend to be Tack Drivers!
Thanks