As was mentioned before there are a lot of options out there and bullet choice will matter even more than chambering I think. I was using a 243 win as my coyote rifle for a while and had great luck shooting factory loaded 100 grain sp's. Very little fur damage and dead right there coyotes. I have never tried a 1000yd shot on a dog with a 243 but I imagine that with a good handload and a lot more skill than I posses a guy could make that happen.
I did shoot 1 coyote with a 308 and it dang near ripped him in two. That was a 150 SP at 80 yards though. But I've always wondered if a 30 cal FMJ would reliably drop a coyote before he could run off too far...