Will i become a criminal?

So I finished my university this year and will be graduating next month, I was thinking of doing some part time job and a little travel until my visa expire which is mid December before returning to my country but this morning my uni sent me an email saying that " If you continue your part-time job after the end of this semester, September 20, you will be considered an illegal worker. If you remain in Japan for more than one month, you will be in Japan illegally and the Immigration Bureau may detain you." which blow up my whole plan cause i though it would be fine as long as i don't overstay my visa. It's too late to apply for new visa now and I dont understand why I can't work or stay in Japan when my visa is still valid. Any thought or recommendation would be highly appreciated. Thank you.

Edit: I ask my university career office and they said as I already informed them that my future plan is going to graduate school in another country they cant provide me the document to apply for a "job hunting" visa without a valid reason ( traveling clearly doesn't seem valid to them), I also went to the city hall to ask and they said I can stay in Japan even after graduating as long as I dont overstay my visa. Didn't ask about the part time job though.