The adjective interschool is good for describing interactions between different schools. An interschool library program makes it possible for students to borrow books from several different school libraries.
You'll most often find the word interschool describing athletic competitions, like interschool lacrosse tournaments. It might also pop up in academic rivalries, like an interschool math team challenge, or more cooperative school projects, like an interschool Chinese club. Inter- is a Latin prefix that means "between" or "among."