As well as the first tier international brand car rental companies, like Hertz, Avis, Budget, Eurocar there locally owned second tier car rentals. Some have on airport depots. Some have new cars. Others can have cars that tend to be ~5 year old (or more) Japanese cars. 10's of thousand of used cars, pickups & trucks are imported into NZ every year from Japan.

http://www.apexrentals.co.nz/
https://www.acerentalcars.co.nz/
https://www.jucy.co.nz/
http://www.rentalcar.co.nz/
http://www.gorentals.co.nz/
https://www.omegarentalcars.com/

Note some rental car companies do not allow you take a car across Cook Strait (North Island to South Island). They get you to drop off in Wellington and pick up a new rental in Picton (or vis versa).