I guess (and you could check this with INZ) you could apply for a further visa of the same kind as you now hold
https://www.immigration.govt.nz/new-...-visa#criteria, before the present one expires, to bridge the gap while you're applying for your Residence, because this application would mean INZ could issue you with an interim visa to cover processing time. Being the same kind of temporary visa you would be applying for, the interim visa would allow you to continue working.
https://www.immigration.govt.nz/new-...isa-conditions (They can't issue interim visas for residence applications.)