on the north island where hamilton is there is pigs, goats, fallow deer, red deer, sika deer, rusa deer, sambar deer and more i can't remember...
south island has pigs, goats, fallow, red, wapiti and whitetail deer, tahr, chamois plus more again...
NZ is a sportsmans paradise!
plenty of info here-http://www.doc.govt.nz/
there is more more public land to hunt than you can cover in a lifetime.
contact shankspony on here, he lives on the north island. i'm sure he would be happy to give you info from a local perspective.