Student visa to work visa

Hi you all.

I read a lot of things online about this but couldn't find the correct/ right or 💯 sure as an answer.

I just graduated from University here in Japan. I did a job hunting since last December but couldn't secure a position yet. Now, I'm still waiting for a company's answer. I really hope it's gonna be yes. My residence status is valid till October but I've just graduated this March.

However, what will happen if I only secure a job in late April without changing a residence status but still continuing job hunting till that time and they offer me a work visa sponsorship. And then, I change my residence status at the beginning of May? Is it risky or what? And is it also legit that we can't do any part time jobs anymore once we rae graduated even if our residence status is still valid for a fee months?

Anyone knows any immigration website that can answer all of these concerns of mine? Really appreciate it.