You can send a resume and use an agency to find a private school job. However, using your network is the best way to find a private school job. 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. Firm up your 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 you've been doing? Or are you seeking different responsibilities, such as being an administrator? You need to have your goals clearly understood. That will be invaluable during the interview process when the 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 advance your career by networking.