I don't think that the fact she was born in NZ is significant. It depends on how many others have tested and where they live, as to what matches you get. Any match may be generations ago, prior to the ancestors moving to NZ, and you still may not find a connection if you don't have a paper trail to follow.
As you know some of the mother's family history, you could consider a Familyfinder test at FTNDA
https://www.familytreedna.com/products.aspxAn advantage of this is that it is one of the cheaper tests, and it gives matches in the second to fourth cousin range which may be useful for your purposes. The disadvantage is that both her paternal and maternal matches will be mixed up, so you won't know which side of the family they come from. As you know her mother's family history you may be able to work out which is which.
You are likely to get a lot of American matches simply because more Americans take the test.
I found the test to be accurate as I have just found a match with a second cousin who recently tested. We had been in touch years ago and already established a paper trail.
You may find the same recurring surname in her matches, which could lead you doing some local research. Some people get good results and useful matches.
If there are any family still living who may be able to give you some clues about the identity of the father, even rumours, I think you should ask them. If they seem reluctant to reveal any secrets, telling them you are taking a DNA test, might spur them on to tell you what they know.
See what advice you get from others, and good luck.
