I've been away from the area for two years but I'm quite sure there are no longer any options to private hire from Queenstown.
As mentioned above, the Wakatipu Aero Club has moved to a grass strip just south of the Kingston township. I don't know what aircraft are available these days. Suffice it to say that the club is a shadow of its former self because of the mistreatment it suffered at the hands of the Queenstown Airport Corporation which looks upon GA as a nuisance.
There may be opportunities to fly from Wanaka. There used to be several training organisations there but I'm not up to date with the present situation.