How can an E-2 visa holder obtain permission to work part-time?

Hello, I am a foreigner with an E-2 visa. I am currently working full-time in Gangnam, and I have already received permission from my visa sponsor. However, I also want to work part-time, so I am curious about how to report this to the immigration office. Should I directly write a permission letter, get my supervisor's signature, and bring it to the immigration office along with the part-time contract? Or are there any available online forms or templates that I can use? I am planning to contact the immigration office already, but I would like to hear advice from those who have gone through this process. Any information would be greatly appreciated!