I'm late to the conversation too. We just got back from Pigeon Forge with all three boys, spouses, and grandkids.
Red I don't know where you stand with that fence but I agree electricity is the best answer. I use electric netting but if I were doing it again I'd probably go with an electric fence in conjunction with a 2x4 welded wire fence. I got my netting from Premier1 and find them really helpful on technical questions when I call.
For electricity to work you need a hot wire and a ground. With the netting the horizontal wires are hot and the ground is your ground. With an electric fence with mesh fence, the hot wires are hot and the fence or the ground is the ground. The netting does not work in snow. I'm not sure if it's because a wet snow shorts it out or a dry snow insulates the critter from the ground. When I asked Premier1 if netting works in snow the lady just said "no", I did not ask why. I imagine an electric fence works well in the snow as long as the lower hot wire is not grounded. Maybe someone that uses fencing can comment on that.
That doesn't mean the netting is worthless in snow. What normally happens with a coyote or about any other critter that can jump or climb over it or dig under it, they investigate the fence first. They typically use their nose or tongue. Somebody on the chicken forum said they bait their hot wire with peanut butter to make sure they test it with their tongue. A critter may test it once or twice but once they are bitten they tend to not get anywhere around it again. So even when it is shorted out in snow or the power is off it is still a deterrent. But new critters that haven't tested it are being born and weaned all the time so it's better for it to remain hot. My biggest problem is not the wild animals around here but has been the dogs that people abandon out here. I'll admit there is a certain satisfaction in hearing a dog yelp and see it running away, unhurt but unlikely to return. Of course I'd prefer it to be the owner that abandoned them but you can't have everything.
My netting system runs on 7000+ volts. The tester only goes up to 7000 volts. It will get your attention if you touch it but I've touched it a few times. The danger is not the volts, it's amps. There is a built in safety system too, it's not a steady current but it pulses. I think there are something like 50 pulses a minute. If it were a steady current it could kill but since it pulses you and the critters can turn loose. Trust me, your reflexes will cause you to turn it loose. A critter's fur or a chicken's feathers will insulate them from the current, but if they touch the hot wires and the ground at the same time with an uninsulated part, like their feet, tongue, nose, comb, or wattles, they are not insulated. I've seen chickens get shocked. They jump back, squawk, and resume feeding. My baby chicks can walk through the netting until they are maybe 6 to 7 weeks old. Their down and feathers insulate them.
My netting has killed a few critters. if something gets caught or is too slow to turn loose the constant pulsing paralyses it and can eventually kill it. I've had three turtles get up against it. Two eventually recovered and walked away when I released them (fun with snapping turtles) but one was caught long enough to die. I've seen a few frogs get caught and die. Once a 5' black racer got caught, I guess they don't back up well. I hated it that it died. And once an opossum got tangled up. The electricity is not what killed it.
My netting is about 48" high. The chickens can easily fly over that if they want to, but mine generally don't. I raise cockerels and pullets with the flock. I used to have a lot of cockerels fly out before I learned a couple of tricks. The cockerels would get in their little fights and the loser would get trapped against the fence. When panic hit they go vertical to get away and land outside the fence. Of course they don't know enough to fly back inside, though you can see that they want back home. I now make sure my corners aren't really sharp. 90 degree corners are OK but flatter is better. I've also learned to avoid long narrow corridors. I once configured it with a 15' wide corridor to reach a wider space further out and two or three cockerels were getting out daily. It didn't take long for me to reconfigure that into a wide area.
One of the big issues with electricity is that vegetation can ground it out. Grass and weeds will grow up In it and, when wet, short it out. Another issue I have with my netting that a fence probably wouldn't have is that dead leaves or cut grass can get washed up against it when I have a frog-strangling gully-washer. That shorts it out. Weed eaters are murder on electric netting. That means when the grass and weeds get tall enough to short it out you need to take it down and mow. You might put It back up where it was or mow a different area first to move it to. How often you have to do that depends on how fast the vegetation grows which depends on the of year and how much rain you get. I can take mine down, mow, and put it back up by myself in two hours or less. It would go a lot easier with a second person. To me that's the biggest disadvantage to the netting. You should be able to set an electric fence up so you could weed eat.
My chicken density is low enough that the area inside the netting is not a barren wasteland. The chickens eat certain vegetation and let the rest grow. That means a few times a year I have to mow inside the netting to knock down the bad stuff and let the good stuff grow. Yu need a way to get a mower inside the fenced in area, netting or fence.
I've had my netting for about five years. In that time I've lost two chickens, one to an owl and one to a hawk. In the three years before I got the netting I lost two chickens to wild animals, but I lost eight one time and five the next to dogs people had abandoned. I consider my netting to be a good investment.
Thinking about the saga with your neighbors, I don't think shooting the coyote is an option. Wouldn't they love to get that deputy back out there? Trapping it is possibly an option but then you have to dispose of the darn thing. You might want to investigate whether or not there are legal restrictions about electric netting or fences, some jurisdictions have them.
That's enough typing this morning. If you have any specific questions I'll be glad to try to answer, especially on the netting. Others on here have better experience with fencing.