In special circumstance, you might plan to apply D10 visa, which allows you to stay in Korea while you are looking for new employment. Korean immigration Office generally requires the following documents.
Checklist to apply D10 from E2
Checklist to apply E2 from D10
*Additional documents may be requested from the Immigration office.
**Application form, an Application form for Employer Change, Confirmation letter for the Accommodation and E2 Certificated Hospital List can be found at the Immigration Office or downloaded from the website: http://www.hikorea.go.kr/.
It takes about three weeks for the Immigration Office to change your new ARC from E2 to D10 visa. It also takes the same time as from D10 to E2. It is illegal to work at a school without having an ARC with E2 status; there shall be a huge penalty. Therefore, D10 visa holder has to apply three weeks before the contract start date (For example, if your start date is September 1st, 2018, Visa application for change of sojourn status must be applied by August 7th, 2018).
So be careful when you apply for D10 visa, and get consulting from professional recruiters at Korvia consulting first or inquire your concern on Korvia Facebook group.