One topic that my book, How to Get a University Job in South Korea: The English Teaching Job of Your Dreams didn't really touch on but perhaps should of is what to do if your timing is bad when you want to work at a South Korean University. The semesters start in September and March, with March being the start of the academic year. A recent trend in the past few years is that schools have started to try to hire most of their English teachers for a March start, when in the past the hiring has generally been divided pretty equally between the two (that said, I got both my jobs as a September start).
Anyway, what should you do if your current contract does not finish at an ideal time, such as in October or April. The best solution would be to stick at your current job until you can finish at an ideal time. This will probably involve asking your employer to extend your contract, but not for a full-year, perhaps only 3 or 4 months. In my experience, most employers will be willing to do this if you are a good employee since it saves them the hassle (and expense) of hiring a new person for a bit longer. The additional bonus of this one is that you can hopefully transfer your visa from your old employer to your new one, which is much easier than getting a new visa. If you aren't a good employee, of course nobody will go out of their way to help you.
The next best solution would be to finish your current contract, but then find some sort of temporary employment such as at a summer or winter camp. Or, you could study Korean for a few months and switch to a student visa. It's kind of unclear whether or not you'd have to submit new teaching English paperwork for either of these options. Perhaps give the immigration hotline a call to find out the details.
If you only have a month or two between your old contract finishing and the new uni one starting, you could switch to a D-10, "looking for work visa" in which case you probably wouldn't have to submit new paperwork.
Alternatively, you could also take a few months off to travel or hang out in Korea on a tourist visa (you will probably have to leave and come back). In this case, you will definitely have to submit new paperwork.
The option that you should put out of your head is starting anytime besides September or March. It just isn't possible. I know it maybe doesn't seem fair, but that's just the way things work here. Of course, think about how you would have felt if you took a class in university and the teacher you started with wasn't the one you finished with. That would just be really annoying.