We were lucky that my employer (I had a job offer) paid for a serviced apartment for the first 10 days, allowing us a few days to find a place of our own to spend a couple of months while our container arrived. The place that we found was an apartment that was privately owned but in a block of serviced apartments. We paid around half the price of going direct to the serviced apartments ($50 a night vs $99).
Quest owns several serviced apartment buildings in Wellington CBD (and the rest of the country) and would be a good starting place before you find private accommodation.