You, the person, are the visa holder. The residence visa is your permission to live and work in New Zealand, and it is on record in the INZ computer, and doesn't depend on the label being in a "live" passport. Anyone within NZ who needs to check your visa status (e.g. an employer, a doctor's office, the police) can do so by contacting INZ. The only time you need proof of the visa in the valid passport is if you travel outside NZ and are trying to re-enter the country, because no airline is allowed to let you board a plane without having seen a valid visa in a valid passport.
So if you're not intending to travel outside NZ at the moment, you don't need to apply for an extension of travel conditions. In due course, you will presumably want to apply for Permanent Residence, and when you have fulfilled the conditions for that, you will probably need to send in both your old passport AND your new passport, and the visa label will be put into the new passport.