You can send in a resume and use an agency to find a private school job. But the very best way to find a private school job is to use your network. You do have a network, don't you?
In truth, the private school job-hunting season never ceases. The main window of opportunity is from December through March. Not much happens on either side of those months unless a school has an unexpected vacancy. This is where your network comes in. Let's say you are considering finding a new position for the next school year. You should try to firm up that decision by the end of October or November so that you can begin your active job search.
The First Steps
Check the job boards in the areas you are thinking about moving to. Register with an agency that specializes in placing private school teachers. They will know about openings. That will get you thinking about the possibilities.
Know why you want to make a move. Are you making a lateral move, i.e., moving to another school to do the same thing as you have been doing? Or are you thinking about different responsibilities, such as being an administrator? It is very important for you to have your goals clearly understood. That will be invaluable during the interview process when the inevitable question "Why do you want to leave St. Swithin's and come teach at our school?" is asked.
This brief video offers three tips to