In most districts, HR is in charge of posting jobs. Often the FA director can't even ask HR to post a job, only the campus principal can do that.
If the particular job you're looking for isn't listed, look for a general "high school teacher" position you can apply to. Send your resume to the FA director and the campus principal.
Unfortunately, if you're doing a broad job search, 95% of your effort will be a complete waste. There's no way to know which 5% of your work will actually be effective.