There have been many methods tried, and in all reality none have been totally successful. Coyotes are a species bent on survival, they have adapted to changing envionments, food sources, and human population dynamics. They are a pack animal, and travel in these family units. The problem lies in the ability to reproduce as the population deems fitting. The more you are successful in lowering the population through trapping, hunting , etc, the more the coyotes mate and have litters. Most wild animal species have set, annual mating routines, but the coyote mates at will. If they overpopulate, and food competition hightens, they slow down, When populations fall, they mate more frequently. In essence they are in control of their own existence.
Poisoning has proven to bee the most successful method of control in the past. Western ranchers used the method in the early 20th century to control predation on their herds, but as you have indicated, that doesn't remain a viable option today.
Try to make it as hard for them to find food and shelter. Sometimes the simple change in the daily requirements will make them move on, but I don't see the species ever being totally controlled. They were admired and worshipped by the early Native American peoples due to their intelligence, diligence and survivability. Keep hunting and trapping them,( they do have decent fur value), and maybe they will move to a neighbors area, but you won't iradicate them. Good luck, I hear their song each night from my back porch, and it's always the cry of success. HH