Oceania (Australia, New Zealand & the South Pacific) - lodging in Queenstown area

Hey there FT. I'm going to be in the Queenstown area from March 2 - 7. I don't really have a set itinerary yet, but I know that I want to check out Milford Sound (and maybe stay a night in Te Anau on the way). Since it'll be the tail end of tourist season, would it be better to plan out exactly where I'll be each day and book lodging in advance? Or would I be ok leaving it a bit open-ended and just walking into a hotel/motel/hostel and finding a room? There will be 3 of us traveling together. Thanks for the advice!

I'm not aware of any school holidays or other big events on around that time, so just fronting in a town would most likely be OK. I've travelled a bit in that area and have had no issues just booking a day or two ahead (I don't like to just front in a place at 6PM without something booked - though I've done it a few times).

I suspect walk-ups would be fine, but probably better if confirmation comes from a Kiwi though!