w4z if you have a contract that starts at a private school within 30 days of the end of your public school contract, I'm pretty sure you can just get an extension on your current visa without having to get a new background check.
ncg115: You should only need one (federal level, apostilled!) background check. If you are planning on getting an E2 visa the background check is valid for six months, unless you're transferring from a private school to a public school, in which case you'll need one no older than 3 months.
If I am understanding your question correctly, there is no grace period. If you are applying to a public school position, you have to have all of your documents ready by the document review period (whenever that is). For a private school, you need to send the background check and some other stuff to Korea before you can get an E2 visa.