Do I have to make a reservation at the immigration office when transferring schools?

Hello! I am a foreign teacher who is finishing my first year of teaching. I am preparing to move to a new school in the Gangnam area as my contract ends in May. I am wondering if I need to make a reservation at the immigration office myself during the process of finding a new job, or if the new school will handle this for me. Before starting work at the new school after my contract ends, I would appreciate any advice on the immigration procedures that I need to prepare for myself. If anyone has similar experiences and can share advice or information, I would be grateful! Thank you!