Yes cheapest and easiest way to stay is to get partner sponsored work visa ( thats the one JandM was referring to ) or residence through partnership ( 12 months living together ).
If you are not living together yet but you are in a relationship try to move in with your partner immediately and start collecting partnership evidence like joint bank accounts joint tenancy agreement , joint bills etc
Another option and speaking from my own experience if you have done three months seasonal work in the horticulture or viticulture industry you would be eligible for another 3 months open work visa under the working holidaymaker extension scheme. which you could use to save up more partnership proof and keep working in the meantime .
http://http://www.immigration.govt.nz/migrant/stream/work/hortandvit/workingholidaymakerextensionvisa.htm
Good luck !