Emily, I never did the type of post-contract traveling that you describe. However, you can find all sorts of posts on various ESL-related Korea discussion forums regarding this topic. An overwhelming number of posts inform folks that most school/employers will not cooperate with such plans by teachers, making it basically impossble to use the return flight home, after traveling to other countries (or even within Korea) after completing the contract.

Don't expect your school to cooperate and schedule your return flight home according to your after-contract travel plans.

Good luck... and happy trails...