I would bring every kind of clothes good for lots of different types of weather.
Here's what wildwalks.co.nz says about September weather:
Winter does not last long in New Zealand and September sees a major difference in the weather. By the end of the month lambing is in full swing and flowers are blooming in the towns. While it is still good skiing the contrast between the mountains and the valleys is dramatic. It can be full on winter at the ski areas with shorts and tee shirt weather in town. Days are noticeably longer and the sun begins to get some heat back into it. Spring snow conditions are interrupted with new snowfalls, often the heaviest of the season.
Climbing and Mountaineering
While spring is approaching in the lower elevations, winter conditions still exist in the high mountains. Heavy snow can be expected. Good access to climbs exists due to snow and if freeze thaw conditions exist, good climbing can be had during fine spells. Watch for falling ice immediately after storms, as a rule it is best to wait a day or two after bad weather to let conditions stabilize. The Matukituki Valley has some of the best crag climbing in New Zealand. Local shops have guide books for the area.
The mainly dry winter conditions which keep rivers and streams levels low now starts to change dramatically. Rain or warm weather will bring down snow melt, particularly in the afternoons. Safety on River crossings is a major concern. Rivers can rise to a point where crossings that were safe in the morning are dangerous in the afternoon. Main tracks will have bridges on most of the major rivers but even small side streams can be an obstacle with any extended rain. Check with DOC to find out track and hut conditions.http://www.wildwalks.co.nz/hiking_calendar.htm#sep