Is not just the South Island that gets cold in winter. Inland North Island, Wellington can all get pretty chilly.
Really the only places that are "mostly immune" to true cold weather and/or wind are the West Coast of the South Island, Northland and the Bay of Plenty. Everywhere else is at least somewhat vulnerable to frost and/or cold southerlies.
I know you say you don't like wool, but really cotton is pretty rubbish in the NZ climate. If it gets wet (and it probably will) then you get cold, simple as that. Whereas wool will keep you warm even if its wet. It might be worth checking out some merino wool over here, it feels quite different to conventional wool, no itchiness. And Possum/merino blends are another option, but expensive.