The visitors hit out after lunch to declare on 352 for nine before Tom Latham and Jeet Raval edged the hosts to 42 without loss.
Bad light brought the day to an early end, and now New Zealand face their highest Test run chase ever, 340 short of a formidable 382-run target.
The hosts are hopeful for a rare series victory against England if they can hang on for a draw after winning the first match in the two-Test series.
“It would be huge,” said batting coach Craig McMillan of what would be the New Zealand’s first series win over England since 1999.
“Series wins have been few and far between for New Zealand Test sides and in terms of history it will be very special.”
England meanwhile see hope for ending a 12-match winless streak away from home.
“There’s definitely enough in the pitch for us to take the wickets,” England batsman-wicketkeeper Jonny Bairstow said. “We’ve beaten the bat consistently.”