I don’t know if anyone’s heard horror stories about teaching at English hagwons (private schools/secondary schools in Korea), but they’re definitely out there.
A couple of reliable schools are CDI/Chungdahm (where I work) or YBM. Basically, a good option is anywhere franchised, where risk of the school having money problems is low. Once you get here, you might be able to learn about smaller hagwons that are also stable. But no matter where you work, you’ll start around 3 or 4 and work until around 10 or 11, since your students go to primary/public school during the day.
The alternative to a hagwon is a public school. You’d either go through the Ministry of Education’s English Program in Korea (EPIK) or it’s Seoul-based program, the Seoul Metropolitan Office of Education (SMOE). The typical public school teacher works from around 8-4 (I think?), has a very secure job, but is paid a little less than a hagwon teacher. But ultimately, both have their pros and cons. Just don’t sign up with the underpants gnomes of Hagwons, of which there are plenty: ‘HAY GAIZ! I GOT GREAT IDEA! 1) WE OPEN UP KOREA SCHOOL THAT TEACH ENGLISH! 2) ??? 3) N MAKE TONZ OF MONNIEZ!’