You can stay here and don't need to go outside of US....if you want to go outside and you don't have valid visa or visa ending soon you may get in trouble...

So, 60 days from OPT expiration you have to do the sevis transfer to different college and your program should start with 90 days of the transfer.

College transfer time:till Oct 30th.
Semester start time:Jan 30th.

GL.