I worked there about 5 years ago, and I know things have changed since I left. But … as far as I know, all you really have to do if you are applying inside Korea is find a hogwan you like and then give your particulars to the director/boss. He’ll run it through the contacts he has provincially and/or in Seoul, and you should have a temporary visa, ID card and work papers IF you meet the criteria for working as an English teacher in Korea and you have original documentation, things which will last until you get the real papers after about 4 weeks. So, no need for a visa run. Best of luck … try Daegu or Changwon, both of which are not your typical Korean cities but which have their own innate charms.