You’ll lose your neighbors’ goodwill fairly quickly if you give them a bunch of stuff to do. They can help tremendously if they just crack open doors of garages/outbuildings and text you if they see your cat. If they live close by, ask if you can have a quick look around their yard in just case. Be neighborly and diplomatic. Try not to overstay your welcome. A search goes south quickly when neighbors get annoyed. When they say no, believe them, and don’t persist. There’s always another way.