Basically, no. A student job is reserved for students. As soon as you have finished your studies and no longer have any commitments to your school, you are no longer a student - so you cannot work a student job anymore. However, there is one exception.
If you finish your studies in June
After you have graduated in June, you can still do a real holiday job in the summer holiday. If you get your diploma in June, the NSSO accepts that until 30 September of that year you can still work and pay reduced social security contributions.
If at the end of your student job you immediately start working for the same employer as an ordinary employee, the nature of your regular job should clearly differ from that of your student job. It is not the intention that your employer use the student job as a trial period for the regular job.