The ESL industry in Korea hires year round. You will seldom ever see much varying ebb and flow of job ads and openings at any particular time. There are over 5000 hagwons of which do not operate on fixed seasonal schedules like public schools and universities. As for public schools and university positions, you will often see some increase in openings in the summer, gearing up for the training in advance of fall classes or you can also see them run ads in the winter for semesters beginning in March. Regardless of that, you will still often see many universities and public schools hiring at different times of the year due to different reasons, such as a needed increase of teaching staff due to class restructuring or perhaps a teacher leaves mid-contract, etc...
Also, Korea always has english camps. These are summer and winter english immersion camps running for a 4-5 weeks during the summer and winter breaks in the school schedule. Hiring for these usually begins 2-3 months in advance of the camp.
If you have all of your documents in order and do not have many job, location, or particular preferences, you could find yourself working in, as early as, 3 weeks or so. The most common delays that usually impede that are when teachers don't have their degree, transcripts, or passports ready, so don't delay on getting yourself organize, even if you don't expect to be available for work in a few month's time.
