Generally speaking in NZ you have to stay on the legal tracks as well, though in the South Island especially there are quite a lot of unformed roads running over private land which vehicles are legally allowed access, though there can be issues where the established track doesn’t follow the defined legal access, or geography means the defined legal access can’t be followed.
https://www.adventureridingnz.co.nz/gps-tracks-rides/getting-access-to-private-farmland/
Keep an eye out for local ‘trail’ rides and enduro events, over summer most rural areas in NZ have an event on almost every weekend, often local farmers group together to provide farm access to fundraise for their local school and the like.
Worsleys Track would be a good place to start, only a little way out of Chch.
https://www.adventureridingnz.co.nz/gps-tracks-rides/south-island-tracks/worsleys-track/
Also check out this from the local council