There are teaching jobs in New Zealand, most are either TEFL or volunteer positions and can be found at private language schools, religious organizations, non-profit programs that work with immigrants, libraries, community colleges and summer English immersion camps for foreign children.
How to Get a Teaching Job in New Zealand
In English-speaking countries that you are not a citizen of, opportunities will be fewer since you would be in competition with local native-speakers. Most English speaking countries like New Zealand are quite protective of their local work force and make obtaining work permits quite challenging. But you can find positions throughout the country in places like Auckland, Wellington, Queenstown and Christchurch.