I ha a problem with coyotes, my good friends dad has been hunting them his whole life. He told me the best way to get them is to trap them. They are a lot smarter than what most people think.
Now this is just an option if you do decide to trap it.
He told me the best way to trap them is a leg hold trap. You'll have to check if they are allowed in your area.
Making sure they can't smell you is probably the most important part. You'll have to wash your clothes with scent less detergent and put them in a bag and leave them outside over night. I'd also spray the clothes with a scent mask, used for hunting. You need rubber gloves and tall boots and make sure both of those have been washed and sprayed. Spray the trap and leave it outside for a few days to get any other smells off.
He digs a hole about 8-10" deep at a 45 degree angle. He usually uses chicken or beef or any other kind of meat in his freezer. He puts the meat at the bottom of the hole and sets the leg trap in front of it, and puts dirt and leaves over it so its hidden. The coyote steps in the trap, trying to get the meat and the trap holds its leg.
My friends dad shoots them then, but with neighbors close by that doesn't sound like an option for you. I'd just call animal control or the police and they'll take care of the animal.
It's up to you really and the regulations of where you live.
But regardless, if you want to catch it yourself, your best bet is to trap it.
However I just can't be sure there isn't more harm in those woods just waiting to get to him therefor he is now being reformed into an indoor kitty whom desperately needs a friend.
