Like Kiwi and others have posted, New Zealand has plenty of DIY open hunting. For my probably once in a lifetime trip to NZ, I chose to hunt with an outfitter. My Chamois and Tahr hunts were free range, although I did use a helicopter to get me up into the high peaks, but my Red Stag and Fallow Deer hunts were in high fence enclosures, which with this outfitter were many thousand acres, much like many of the South African properties.
I have no trouble hunting high fence properties as long as the property is at least a section (640 acres) or more in size that the animals have plenty of area to get away or hide from hunters. Its really no different than farms or ranches here at home where a hunter gets permission to hunt there. The animals are free to leave the property, but the hunter is not.
Anyway, back to my New Zealand hunt, after my first day of Stag hunting, I knew that this was probably going to be my only hunt in NZ, so I upgraded my hunt to one for a gold medal Stag, and ended up with this Stag of a lifetime...