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 5,000 hagwons, few if any of which operate on fixed seasonal schedules as public schools and universities do. As for those public school 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 which run 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.
When do schools advertise?
If looking at ESL job boards, you'll see that 99% of the jobs are for starting dates ranging from ASAP to within the next 2 months. Very seldom will you ever see job ads for start dates that are 3 months down the road. And almost never will you see job ads for 6 months to 1 year down the road. One reason, is that most jobs are currently held by English teachers who don't often make up their minds if they want to renew their contract with the school until the final 3 months. And even if they've decided before then, Korean directors tend to not start looking for new teachers until 2 months before the position begins. It's a fast moving market and if you plan to come to work in Korea 6 months later or more, don't be surprised or discouraged if your inbox has no job offers in it.
