Certainly anything up on the three mountains Ruapehu, Tongariro, Ngauruhoe, the alpine crossing (which is part of the round the mountain walk) is going to require iceaxe and crampons. Spring snow in New Zealand is very variable and is often quite wet and icy. AS noted above spring often has a lot of wind. The lower level tracks in the bush would tend to have less snow.
Other options nearby are the Kaimanawas and Kawekas to the east, Kaimais to the north, and Ruahines to the south. All these options except the Kaimais would have the tops covered in snow, but the valleys are likely to have less/little snow. These areas are much less developed than Tongariro NP, the tracks are rougher/more difficult, the huts smaller/less well equipped but cheaper, also far less people around. Transport to these ranges is more difficult too, when compared to the number of shuttles/buses available for Tongariro NP.
Long coastal walks are pretty rear in NZ. There's 90 mile beach (3 days) about 8 hours drive from Tongariro, or the Heaphy track or Abel Tasman in the South Island.