Immigration Act 2009, ss 14(2), 20, 71(1), 79(1)
If you’re in New Zealand and you let your visa expire, you’ll be here unlawfully and won’t have a right to apply for a new visa. So, if you’re in New Zealand under a current visa and want to stay longer than this visa will allow, you need to make sure you apply for a new visa before the current one expires, and that you apply early enough for Immigration New Zealand to process your application before the current visa expires.
The fact that you’ve simply applied for a new visa doesn’t of itself give you a right to be in New Zealand. However, Immigration New Zealand can decide to grant you an Interim Visa so that you keep your legal immigration status while they’re considering your application.
If you’re here unlawfully, one option is to leave New Zealand voluntarily and access the visa system from overseas. Leaving New Zealand “resets” the system for you, allowing you to apply for a visa once again. If, however, you stay here until Immigration NZ forcibly deports you, you’ll face a ban on returning to New Zealand for a certain number of years (see “Accessing the immigration system: Who can apply for a visa and who can’t”).
However, it’s not totally impossible to get a new visa from within New Zealand once you’re here unlawfully, because Immigration NZ has a broad power (a “discretion”) to grant you a new visa in this situation. However, you have no right to apply for the visa in these cases and have Immigration NZ properly consider your application. They can simply choose to ignore your request for a visa if they want to, without even considering it. It’s different when you apply while you’re here legally, as Immigration NZ then has a legal duty to at least consider your application.