If you are familiar with the local private schools (already know where they are, etc.), you can always go by and ask for an employment application.
That's what people do around here. We have a file of applications and will be hiring 3 new teachers next year...young school having major growing pains.
After you give them your application, check back in later and let them know you're still interested.
I got my position because I was working part time at the school (answering the phone) and was in the building the day a teacher quit. She quit on Friday and I started on Monday.
We also keep a sub list of those willing to do the job, but the list is short in our area. Our school board has the last say, but they haven't refused anybody, as far as I know. We even have a listing of who not
to have sub.