There are no true International schools in New Zealand so your best bet is TES or the NZ version, Edgazette. It’s also free like TES. Some schools offer the igsce and/or the IB diploma as an alternative to the NZ NCEA system. Jobs start to be advertised now for the new academic year that starts the end of Jan. that’s why you’re seeing them in TES.
I taught there for 12 years before my current job in Asia. Absolutely loved it and plan to return. It’s a great country. It is geographically isolated though which makes travel expensive with very long flights.

It has the full range of schools and students just like the UK/USA - rich, poor, religious, non-religious, private and state, huge inner city and small rural schools, etc.
I’d highly recommend it.