It may not be so easy I am afraid. I assume you are travelling down from India for a holiday. If you have flown in the South Island before you may already have contacts in the region.
Without the contacts and possibly any local knowledge or experience you may be perceived as too much of a risk. Most private owners would love to have the hours flown off their machines but the insurance issues would deter them from hiring to experienced locals even.
I am not sure what airfields you have in mind but travelling to and fro the townships/cities is often not always easy either. Ashburton marginally better than Timaru for example.
You may have better luck travelling about by car or campervan and then doing daytrip flights.
The whole of NZ goes on holiday during January so it can be a bit of a crazy month in many respects.