It is extraordinarily difficult to move from a relatively low paying teaching job (such as in the NYC DOE) to a high paying job in Westchester. There is far, far more labor supply than there is demand. If you don't get in when you're young (and therefore cheap), it's most likely not going to happen.

When I was growing up my father worked in an "elite" Westchester private school and my mother worked in the NYC DOE. The job fulfillment, administrative support and overall ease of working was like night and day between them.