No, you have a month after your visa is done. During that time, get a D10. It is good for 6 months. It is a looking for work visa. You can stay in a hotel, motel, goshiwons? Stay in Seoul for a bit, then go down and find one in Busan? Spend 2 or 3 months looking for a quality job. Plus have a break in between.
Is the concept of opportunity cost lost on you?
There's no opportunity cost if the job pays Korean minimum wage and you take it out of desperation. Take the time to find something better and higher paying.
If you spend two months without work looking for a job, you've lost two months' worth of income, even if it is minimum wage.
You can always look for a new job while working at your current one. It's not like there's a huge discrepancy between hogwons.
Changing from Public to Hakwon is akin to moving from Seoul to Seattle.Step-down transformer.
Down?Teach whatever lessons you want instead of forced materials? Have small classes of high level students instead of huge classes of waaay vavied level kids? Live in a city never either too hot or too cold in anyone's books? Give me an academy in Seattle over a public school in Seoul any day. A step DOWN? Hashtag run down! (Getting a green card is another thing)
Van should come work in China and make some real money
How's the English teaching climate in China? I have a friend who often talks about going. He hasn't made the jump yet but I'm sure he will. I'd feel much better knowing he'd be able to land on his feet over there.