Mid winter north island tramps?

A couple of mates have recently caught the tramping bug and are keen to do a 2-6 day tramp in the next month or two. We all recently live on the north island, so aren't that familiar with nice spots. As stated, they're reasonably new trampers, so avoiding anything technical (though all would be fine with low grade scrambles), likely to be snowy, or with too much off-trail navigation. Lake Waikeremoana was suggested - and perhaps mid-winter is a good time to try it with presumably less foot traffic around (i'm not usually a fan of the groomed and heavy traffic of some of the great walks). Huts aren't a necessity, but if the weather is poor, having them available will probably help morale! But what other accessible, scenic tramps might you folks suggest? Waikeremoana is probably as far north as we want to go (they both live in Wellington).