Go out and present yourself at the institutes and you should be able to pick up plenty of hours. You will need a presentable C.V. or Resume as the Americans call them and your appearance should be as smart as possible. Useful additions might include such things as photos of you teaching, and some teachers even supply videos of their previous classes. If your institute has a camcorder among its facilities then try taping a few of your classes, you never know when they might come in useful.
Employers still find earrings, long hair etc.
Some of the Hakwans have western academic directors and these are the ones to head for as the foreigners are far less judgmental and much more reliable when it comes to getting paid. Korean directors are sometimes quite bigoted and will not hire blacks, non Korean Asians or westerners without a North American accent. Be prepared for some serious snubbing. Globalisation is the current number one buzzword in Korea at the moment and they feel that the key to this is learning English. They open their doors, hire the first foreigner they can find, advertise, teach for a month or so, and then close.