I've heard that their breeding season and calving season are some of the best times to hunt them. Anyone?
What is the 'best' time of year to hunt them is going to vary a lot between hunters, circumstance, and their methods/styles of hunting. Obviously, when calving is going on won't matter one iota to the hunter who is hunting a property without cattle.
I've heard that when the cows are calving the coyotes seem to always be nearby.
Coyotes are always nearby when out in the country. The only questions are just how close they are and if you can see them.