I'm going to work under the assumption that you have already started to apply for every job under the sun so far. If you haven't....get started, right now, on applying for every job under the sun (unless you have a valid reason to limit your job search).

Without trying to sound snarky, when should you start to panic? On the last day of TBA. As a new grad, you have an advantage that us vets don't...the 45 day rule doesn't apply to you this summer. A job can become available on July 15, and you don't have to sweat out the possibility of your current district not releasing you from your contract to take that new job.