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Your new employer needs to obtain an approved LMIA so that you can apply for a new closed work permit.
Scylla, if the potential new employer can obtain an appoved LMIA, can the Closed Work Permit holder change to an Open Work Permit with this employer full time job offer?
 
Scylla, if the potential new employer can obtain an appoved LMIA, can the Closed Work Permit holder change to an Open Work Permit with this employer full time job offer?

No.

All you can do is apply for a new closed work permit tied to the new employer.