I'm currently working full-time on an E-7-1 visa?

On an E-7 visa, including the E-7-1 subcategory, your work authorization is tied specifically to the employer who sponsored your visa and the occupation listed on your permit. Taking on a second job without additional immigration approval is technically a violation of your visa conditions, even if it is part-time and in the same field.

That said, there is a formal process to get this approved. You can apply at your local immigration office for permission to engage in activities outside your designated status, called 체류자격 외 활동허가. Approval is more likely when the additional work is in the same or closely related field as your main job. In most cases the immigration office will want to see that your primary employer is aware and not opposed to the arrangement, though a formal letter is not always required.

I went through this process myself and it is fairly straightforward as long as your employer does not object. Having an honest conversation with your main employer first is the smart move, both because immigration officers sometimes contact employers during review and because undisclosed second employment can complicate your visa renewal later. The Hi Korea portal at hikorea.go.kr has the application form and a checklist of required documents.