The Grey District, Buller District and Nelson Tasman are among the regions in the South Island to have declared a state of emergency, as has Taranaki in the North Island.
National carrier Air New Zealand had cancelled several flights from Wellington since Tuesday afternoon and was working through the backlog on Wednesday.
Medical supplies have been picked up in Nelson and flown to Golden Bay in the helicopter that's taking two cabinet ministers around parts of the upper South Island hit by former cyclone Gita.
Flood waters remain in residential areas, where people have begun the task of cleaning up.
Several tourists reportedly had to be rescued by authorities after becoming stranded on roads near Whataroa, the New Zealand Herald reports.
Residents have been urged to stay inside and not to travel unless absolutely necessary.
A staggering 256 millimetres of rain fell in the Kaikoura Ranges in just 18 hours, and of that, 53.6mm of that fell in one hour, making it the wettest hour ever recorded in New Zealand.
Nelson City Council communications manager Paul Shattock told local media the military had been deployed in Takaka and was helping to get people to safety.
The storm system is the remnants of Cyclone Gita, which ripped through the Pacific nation of Tonga last week.
Dozens of schools have been shut and roads closed in the South Island as the storm made landfall on Tuesday.
The storm caused major slips on the Takaka Hill highway, the only road access in the area, which could take days to clear.