Looks like an assignment on North and South Islands for a few weeks. what is "in season?".

Knowing that it is late winter, how about fishing? are the trout spring spawning runs on yet?

look forward.

Great time to pick up a spring spiker for meat.
The Red Stags start shedding late August/September.
One of the best times of the year to hunt a trophy Bull Tahr. (Everyone seems to think the rut is the best time for some reason.) Their manes are fantastic in September and often they'll come a little lower in Sept/Oct.
Chamois coats are getting past their best.
Trout have done their thing as far as spawning goes.
There are areas that you can fish for trout in August/September. (Not the prime areas)
Places like west coast odf South Island are open for trout fishing all year round.

Still abit of bird hunting available.

Wallabies in the South Island.

There is hunting and fishing a plenty all year round in New Zealand.

Fishing season for most of the country starts first of October , but Taupo fishes all year round , with some restrictions.

Hunting would likely be limited to meat animals rather than trophy game , but the alpine animals , plus goats and small game are available all year round .Some of the guides/outfitters would be able give more precise information .