I thought I would share my experiences with hammocking in New Zealand, since I'm always happy to read stuff like this prior to travelling.
I spent 6 months in New Zealand recently, travelling first by car and then by bicycle. Due to a postal foul-up, I only had my hammock for the last 3,5 months while travelling on North Island. I stayed strictly on designated camp sites; I never stealthed, although I kept this as an option.
Generally the camp sites in New Zealand were A+. I've never seen more luxurious camp sites anywhere, and I'm afraid I never will. All camp sites had cooking facilities, most had community areas like a place to eat or watch TV. Sometimes there were even games rooms. Everything (including toilets and showers) was usually spotlessly clean. My guess is, that I stayed in 50+ camp sites, and there were only 2 or 3 that were dirty (well, for New Zealand standards). Camp site owners were usually extremely friendly and willing to find me a spot where I could hang my hammock.
I could only use my hammock for around 2 weeks on South Island. During these, I usually had a spot to hang in 1 out of 2 campsites.
On North Island the ratio was better - maybe 3 out of 4. However, if you are planning to do something similar, my advice would be: bring a suspension that is very flexible. Most times, I had to use trees that were VERY far apart (9-12 "steps" - so maybe ~10m). A couple of times the distance was just about enough to hang the hammock. After these experiences, I converted back to webbing, as I couldn't go short enough with whoopies.
I did most of my hanging in the off-season, which means that campsites pretty empty, and I had a good choice of sites. Most of the time, there were only one or two sites per camp site, which would allow hanging. Had I been hammocking in the high season, I would probably have had to arrive early to get a hammocking spot. And I probably would have had to use my tent more often.
Stealthing would have been possible, but often illegal.
On the hikes I did (Fiordlands, Queen Charlotte Track, Abel Tasman, Wanganui River), hanging would not always have been possible, and the DOC people could never tell me beforehand what the "tree situation" was around the designated camp sites.
All in all, I would probably recommend to take a tent in addition to the hammock. Or risk staying in hostels/hotels now and then.
Hope this is helpful.