International students who have graduated from a university or vocational school cannot stay in Japan with the status of residence " Student".
Even though you have graduated, staying in Japan for "3 months or more" with a "Student" status of residence can be a reason for revoking your status of residence.
Job hunting after graduation is also a reason for revocation of the status of residence, as it means that you are not engaged in activities according to your status of residence and you are staying in an attempt to engage in other activities.
Therefore, in principle, you should decide where to work by the time you graduate and change your status of residence according to the job you will get after graduation. ("Change of status of residence")
(However, even if you have an unofficial offer, you cannot be relieved.
It can be the same situation as if you did not get a job, such as canceling an unofficial offer or not being allowed to apply for a residence permit in the first place. )
So what if you haven't decided on a job?
For such "former" international students, the change of status of residence to "specific activities" for the purpose of job hunting is permitted.
This is called "就職活動目的特定活動許可(syuusyokukatsudoumokuteki-tokuteikatsudoukyoka)".
As a general rule, the period of stay of this status of residence is "6 months", and the period of stay can be renewed up to one year after graduation.
In other words, you can stay in Japan for the purpose of job hunting for one year after graduation.
During this time, you can work part-time for 28 hours a week by applying. ("Comprehensive permission for activities outside the qualification")
In addition, as a special case when you get an unofficial offer, you can stay in Japan for a year and a half after graduation.
The following two points are required.
(1) within one year after the offer is decided,
(2) it must be adopted within one and a half years after graduation.
For example, you may graduate from university in September 2020, receive an unofficial offer in August 2021, and join the company in April 2022.
In this case as well, you can work part-time for 28 hours a week.
The "an unofficial offer" here is the so-called "内定(Naitei)".