Here in Florida coyotes often den up in old gopher tortoise holes that they dig out larger to accommodate themselves. By the time the coyotes are done with it, the hole will be about dog sized and at the base there will be a pile of dirt or sand several feet across and a foot or two high. You don't have burrowing tortoises where you live, but you may have animals like badgers or other diggers whose holes coyotes my appropriate for their dens. I presume coyotes everywhere make permanent dens like they do here. Finding them takes a lot of walking, especially though thick. Usually the coyotes will have defined trails that go right to their dens. This is especially so if you can bait them up in a regular location over the course of several months. We use deer carcasses to bait them up and they've made trails going straight to the carcass pile from their den holes. We then set wire snares along the trails and catch them going to and from the carcasses.
Calling them and shooting them is fun, but its not super effective to get their numbers down. They learn fast and after a couple of them put the slip on you in a hunting session, they will become hard to call in. Trapping is your best bet. We use snares here because that's all the state lets us use, but leghold traps are the most efficient way to catch them. Snares work best covering the trails to and from the den and carcass piles. Leghold traps work best with dirt hole sets.
You will never wipe them out as a species, but you can wipe out specific packs and keep them beat back every time a new pack moves in to your area. You just have to stay on top of them.
They kill adult deer with regularity. Don't let anyone convince you otherwise. Where you live there probably isn't a tradition of hunting deer with pack hounds like there is in Florida and southern Georgia. A good pack of deer hounds can get their master 20 deer a year per person easy just by hunting weekends (no bag limits in Florida). A pack of coyotes is infinity better at chasing and killing adult deer than the best pack of walker hounds. A pack probably makes an adult deer kill a couple times a week at least.