Think about your marching program. Depending on what you want to do with your shows, how far ahead of time will you need to secure copyrights for arrangements, hire the arrangers themselves, drill writer, guard writer, any other adjunct marching staff? When do you want the music ready by? The Drill/guard work ready by? Are you going to do a pre-show? Who's doing that? What equipment do you need for the pit that the band doesn't already have? How about any props? Does the school have trailers or trucks in order to haul everything from the band hall to the contest/game and back? Who drives those? How far ahead do you have to file the transportation requests for those? How do the kids get fed? Do the boosters cover that...or is it "every man for themselves"?
It also depends on what size school you are shooting for. If it's a competitive 5A or 6A program, chances are in May a lot of the things I've listed have already been set in motion. You will need to touch base with everyone to make sure things are on schedule. As schools get smaller, less has been done at this point in the year, simply because the majority of small schools aren't contracting out for a lot of things due to budget constraints. To make sure where things are, if you take the job contact the outgoing director and/or the remaining staff to see where things stand.