Great question! There are definitely many opportunities beyond teaching in Japan or Korea.
I would say the next step is to define what you are interested in doing, and then start to research companies that have global offices in those locations of interest.
For example, I used to work at a large consulting firm, Accenture, and they had a few offices in Tokyo and others internationally. Although I personally did not work abroad, there were opportunities to join a team or a project that was located in those countries. I was, however, able to connect with a coworker in Tokyo for an "English/Japanese Language Buddy Program". I met with the coworker virtually to practice my Japanese and help him practice his English. It was a great way to get connected and learn more about the culture.
Another idea, if you are still in school you could try to find study abroad or global internship programs that could fit your interests. I was able to study abroad in Kyoto Japan during my undergraduate degree, and that experience also connected me to an internship in Tokyo for that summer.
Overall, don't give up! You can find something that fits your interests that is also in the country of your choice. I left some suggestions below, and good luck!