Is the best way to become a direct hire, is to undercut your dispatch company?

If let's say, you've been at your school(s) for a year or so and the school(s) really likes you (let's assume you're exceptional). Do you have a chance before the end of your contract to ask the BOE/schools directly if they'd hire you through them and not your dispatch company? Is that a long shot? Has it worked for anyone? Is there a certain time of year to ask to make that chance more likely before they start discussing with the dispatch? I'm assuming you'd have to tell them "don't tell the dispatch I asked" LMAO or you might not get recontracted. From what I've heard, it seems direct hire jobs are usually obtained if you know someone who is leaving and are replacing them. (you're rec'd by them) I'm mostly just curious.