Bottom of the South Island in early May

hump ridge track, you should be able to get that done without crampons, you might miss the big snow dumps then. kepler up at te anau is great, but exposed, and more prone to snow. theres rakiura on stewart island, good quality track kepler is very close to te anau, rakiura is a snort walk from town, theres transport to hump ridge theres also the dusky track , very scenic but very muddy and a few steep passes. you could get on the milford without booking then, you'll need a gas cooker for all but hte hump ridge. you realise invercargill is in the middle of endless miles of relatively flat farmland?, most of these tracks are a reasonable distance away. rakiura is the closest. if you can get north of te anau, not sure if ther are shuttles running in may, then theres far more choice, like the routeburn, my fav, hollyford, , greenstone, caples. mavora, but the shuttles tend to stop going up there at the end of april, you should still be able to get some transport to the divide whre the routeburn, and caples greenstone tracks are.