We have been using Jack Russells instead of cats for a number of years. The cats kept getting killed by coyotes. The only advantage a cat has is it’s climbing ability. However a Jack considers vermin catching it’s mission in life. They will keep after one until it is dead. A Jack on a mission is a sight to behold.
You just need a bigger cat. Even though NH Fish and Game denies it, there have been instances in NH of large cats killing and eating coyotes.