I highly recommend Namgajwa-dong Good Kyunghee Oriental Medical Clinic.

As you can see in the photo, they're open on Sundays and public holidays, which is incredibly helpful when you need urgent care. They also offer nighttime care, which is incredibly helpful. I suddenly felt a bit unwell, so I even visited on Memorial Day, June 6th. They also seem to have an inpatient facility that can accommodate traffic accident patients. While it's a standalone building, it's quite large. If you live nearby, I highly recommend using them for emergencies, such as on public holidays.