I am not sure I understand the exact situation you are in from you postings, but I will try to respond to your question based on what I know.
If you are leaving the employment of the school, which sponsored your current work permit, prior to the expiration of your work permit to go work for the company, you must get a signed permission from the school agreeing to transfer your work permit to the company… when you have the signed permission, you need give this to the company for them to apply for a new work permit. This should all be done prior to you working at the new company since, as you probably know, the work permit only allows you to work for a specific employer, in this case, the school. If you approach the process this way, you will not have to leave the country and re-enter.
If the school is not willing to sign the transfer permission, for whatever reason, you must notify the work permit issuer, Ministry of Justice, when you stop working for the school (by law, the school should also notify the MOJ), leave the country, re-enter, and have the company apply for the work permit from the very beginning. Having the work permit transferred is a much more expeditious process than applying all over again. If you have any questions, you can email me at my email… email@example.com.