Enjoy your time in NZ; it's a beautiful country and the Kiwis are great people. I've spent, all told, about 8 months in NZ over two trips about 8 or 9 years ago.
Both Spencer and Barry have lots of good suggestions which I would give a +1 too.
When I did the Routeburn (May or June, can't exactly remember any more), we combined it with the Greenstone Track. You could also combine it with the Caples Track, I think. Either way, it makes for a semi-loop from the lake outside of Glenorchy, not too far from Queenstown.
The hut system is pretty sweet and makes for luxury camping. It's fun to read the hut journals, visit with others, etc. Plus, when the weather's nasty, you'll be happy to be warm and dry inside the hut with a pot belly stove.
One of my regrets with visiting both times during NZ's winter is I never was able to hike the Milford Track. In fact out of several attempts to just get to Milford Sound, I only actually was able to make it there once due to bad weather and dangerous snow conditions along the road. Anyway, the track looks beautiful and it seems worthy of making the effort to add to your itinerary.
The trip from the town of Fox to Welcome Flat is a fun trip too. Watching the alpenglow from the hot spring is tough to beat! As is a kayak/camp trip through Abel Tasman NP up on the northern area of the south island. If you planned it right, you could probably even kayak one way into the Abel Tasman, then have your kayak picked up by the water shuttle and then hike back out. Best of both worlds!
There was another trip, I wasn't able to do because of winter weather but that a friend did the previous (NZ) summer. I don't remember the exact location, somewhere out of Haast, but he took a helicopter ride into the Siberia Range, was dropped off and hiked out. Just from driving by the general area, I can only assume the scenery along the tramp was stunning. Not sure on the logistics for pulling this off, perhaps one of the resident Kiwis can help, but if I went back I'd look into this area.
Finally, not treking really, but the Catlins area between Dunedin and Invercargill is a really beautiful lonely stretch of coastline worth spending a day or two exploring.