Hi, I really sympathize with your partner. I was in a similar situation, we applied for a visa and I was a dependent. The application was declined (visas had lapsed) and we were dubbed "unlawful". I submitted a Section 61 application but they declined it after 2 weeks.
I hope there is still hope for him.
First you need to submit a letter stating the scenario (when his visa expired, why he overstayed, why he should be granted a visa). Then what i did was I gave them my application form (visitors visa) with a sponsorship form.
Call INZ. They give great (free) advice!
All the best.