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As others have said, June is a hard month to get into the hills. A guide will cost more but increase the chance of success 10 fold than if you just head for the hills by yourself. Drowning in NZ rivers was called "The New Zealand death" back when the first colonisers were rolling in. And Thar country is littered with river crossings to get into their domain.
With one week taken up with hunting, attempting to fit in both the North and South island in, in the remaining two weeks will see you spend most of your time sitting in a car with the steering wheel on the wrong side just driving.
Almost all of our State Highways are pretty much like your secondary minor roads, so quite slow in comparison to what you may think you be able to cover in a day.
I would be tempted to stay in the South Island for most of the trip, then keep 3 days aside for driving to Auckland to fly out, if you really must see the North Island.
You will be spoilt with the low population in the South island (around 1.5 mill now), with the other 4 mill shoved into the North Island, and 2 of those packed into Auckland, making it similar to any other small-mid size city in the world.
Central North Island for volcanoes, possibly Rotorua for mudpools and thats your whack, for three days.
Enjoy
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