the 22lr is good to kill coyotes, that are already in traps!! You need to at least use a 22 mag., the .223 would be even better!
Most any distress call on the market will call a coyote. You need to learn that they will try to come from your downwind side, so you need to set up so that you can see downwind and shoot them before they smell you. Coyote can be called any time of day, but early morning and early evenings will produce best, until it gets really cold, then any time of the day is good.
I'd suggest buying a predator calling video to watch and listen to the sounds that you want to mimic. basically, you need to make distress sounds that have excitement to them. Think about a bunny that is in a death grip of a hawk, or the likes. Call for a minute, or less, maybe a bit more, than sit quietly and watch for a couple mintues. Nothing in view, then do it again. A coyote can show up in 30 seconds, 5 minutes, or 15 minutes, so be ready from the first blow on the call!