I discovered this independently and I'm fleshing it out with details from this OzBargain post https://www.ozbargain.com.au/node/440682 and other sites.
If you are on a 30 day Amazon Prime free trial https://www.cheapies.nz/node/20435 and wait until ~ 3 days before your trial is due to end, a lot of people have been offered the opportunity to extend it for 30 days. In my case I simply had to visit the manage your Prime membership page https://www.amazon.com/mc and there was an intab popup offering me a 30 day extension. I didn't accepted straight away and when I go to it now, the extension offer is still there about on the middle of that page.
Alternatively if this doesn't work for you, try going through the cancellation process and when you get part way through you may be offered a 30 day extension. The cancellation process seems to be designed to offer you something since it's a 2 part process. On the cancellation page, you first click 'End My Benefits' or something similar and they then have another page where they offer you something to keep it.
If you get to the second page of the cancellation process and still no offer, stop, go back to the membership page and try changing to an annual subscription and see if that works.
Unless you plan on keeping Prime, and the advantages in NZ are currently limited, remember to cancel before your extension ends, so you are not charged. (If you signed up for Prime on Prime day your trial is likely about to end.) That said, if you forget to, provided you haven't used your benefits you may be refunded the fee. This says it should be automatic https://www.amazon.com/gp/help/customer/display.html/ref=hp_... but if you have nothing after ~7 days, you can try asking customer service.
As mentioned in the earlier thread, the general suggestion is you can receive a trial every 12 months. I'm not sure how using this extension may affect that nor whether it's offered with every trial or only once.