Family members of a Greek citizen are considered among other categories: the other spouse or partner with whom the Greek citizen has entered into a cohabitation agreement, provided that it took place either in Greece or in front of the Greek consular Authorities abroad. So your partner, after the conclusion of a cohabitation agreement as above, has the right to stay legally in Greece. Furthermore, a New Zealand citizen is not required to obtain a Schengen Visa to enter in Schengen area legally.
What are the options for a New Zealand citizen to stay in Greece for more than a year?
I am a Greek citizen, my partner (not married) is a New Zealand citizen. We live in Australia and are moving to Greece in April to live and work. What are the options for my partner to be able to stay in Greece with me?
You have the option to proceed either with Golden Visa proceedings or with marital proceedings with a Greek citizen.
The easiest option is a Greek work permit for your partner. There are also several other options, such as a Greek Golden Visa.