Contact the Schools
Contact the schools directly and see if a place is available. Phone the admissions department as soon as you can. That assumes, of course, that you know the schools in the area to which you are relocating. But what if you don't? What if you simply don't have time to do all that careful research? The solution is to hire an educational consultant to do the work for you. Consultants know private schools and have the contacts to find a place for a qualified student.
You May Be in Luck If There Are Places
Back to the original question: what if you have missed the deadlines for entry next fall? You probably will be out of luck when it comes to the most competitive schools. But there are plenty of very good schools which have rolling admissions or no fixed admissions deadline. In other words, they admit qualified applicants as long as they have places for them. The other reality is that no school likes to have empty places. But things do happen. Students are forced to drop out of school for all kinds of reasons. Suddenly a place is available at the last minute. So it never hurts to ask.
Check the SSAT site
The SSAT site maintains a list of schools which are currently considering applicants. This SCCA list is very useful as it not only lists schools which are currently considering applicants but also groups these schools by state and by semester of entrance. For example, if you need a school which is located in Massachusetts for fall admissions, you will find that group and search the list of schools.
Finding a place long after the deadlines have passed is nerve-wracking, to say the least. Once you have found a suitable school, you need to apply and take care of all the details. And you have to do all this quickly to ensure that your child gets the place. Again, if you cannot devote the 20-30 man hours this process will take, you might want to consider hiring a professional consultant to handle the applications for you.