How long can I stay in the US if I leave after the first, or second, year?

As the Cordell Hull Foundation Orientation Trainers explain at all the required new-teacher orientations, your J-1 teacher visa status automatically ends on June 30 of the academic school year. You have a 30-day grace period to stay in the US, until July 30.  During the grace period, you can travel or wrap up your affairs, but not work.  And, of course, if you leave the US during these 30 days, you cannot re-enter in J-1 visa status for two full additional years.