If wanting to cross Cook Strait (North Island ~ South Island[the mainland]) check carefully if doing a one way hire.
Some will want you want you to swap cars at Wellington ~ Picton. Others will allow and some do not allow.
With one way hire are always surplus of cars at southern depots. Many people drive south and fly north.
Hence (are others)
Relocation rental cars, campervans and motorhomes | Transfercar
As well as the international majors, are local car rental companies.
Omega Rentals
https://www.apexrentals.co.nz/
Ezi Rentals Was know as A2B Car Rental
Snap rentals
You may pay a premium to rent from the airport. Renting from a city location may be a lot less, for the small cost of a taxi/uber from the airport to the rental car location.
Other smaller rental companies (alphabetical order)
ARentals
Home
Car Hire in New Zealand & Australia | Ace Rental Cars
GO Rentals - Car Rental & Car Hire New Zealand
Car Rental New Zealand - Rent a Car from $49 a day | RaD Car Hire
An others. But avoid the "rent a wreck" very low cost places, with ~10 year old cars.
Some of the lower grade rental car companys can change names.