Once you land at Christchurch or Auckland without a booking, you are basically at the mercy of whatever is left in the lot, and during peak tourist season it is slim pickings. Most folks lock in a decent rent a car well before they even get to New Zealand, sorting the insurance excess and an extra driver online so they can just grab the keys and shoot off to Queenstown without any faff at the desk. You can compare the available motors for your trip dates right here
https://getrentacar.com/en-US/new-zealand .