I could understand working for a defence contractor, for example, would prevent one from haphazardly working other jobs on the side, for security reasons. Let alone for working for something more critical like an intelligence agency. But if you work as a school teacher, the school's reputation is the only factor. But does that give the school a legal right to prevent a teacher from working picking any particular second job? It's purely subjective what damages reputation, after all, aside from gross cases, like working as a thief on the side.
That being said, I don't understand how she would have enough hours in a day, being a teacher. Especially being a teacher in Japan.