I was working late on another project so let a script run for flights in August. Here are the cheapest flights across the motu, from some of the main centres. Sometimes it's Jetstar, sometimes it's AirNZ, but you can use the filters to choose if you have a preference.
If looking ahead to the school holidays, see Air NZ and Jetstar Domestic Flights from $66 One Way (Next School Holidays: Friday 19 Sep - Sun 5 Oct)
Full tables are in the link - it's too big to fit on cheapies!
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I checked Google Flights and some of these tickets can be booked cheaper, but only through (potentially risky) third party vendors/agents such as eDreams. I am seeing some slightly cheaper tickets from eg Auckland to Wellington, Jetstar seems to be doing $63 tickets most days?