Where's Perry Mason when you need him! I bet he could trick a confession out of your neighbor and get it recorded on tape, especially after he sent Paul over to the neighbors garbage to retrieve the leash that he probably pocketed to get rid of the evidence!
If your computer is close to a window overlooking your yard you might be able to just put a webcam on it and have it recording continually or at least when Jack is out in the yard - that might be less expensive option.
I wonder if the police would be willing to talk to your neighbor and explain to him the consequences of getting caught on tape releasing Jack. Even though you don't have any evidence this time, they might be able to scare him off of doing it again.
I hope you get Jack back quickly!