The flattening of the pitch, the confidence of the batters and the support of a burgeoning Sunday crowd seemed to inspire the hosts.
They moved through to 195 for two, losing one wicket after resuming on 90. The deficit is 358.
Ollie Pope advanced from 51 to 84, Joe Root is 35.
Alex Lees was the sole victim, wafting outside off stump with casual footwork off Matt Henry. The snick saw Daryl Mitchell's first slip-catching mojo return after spilling chances off Lees and Pope on the second day. Lees made his highest test score of 67. His previous best was 31.
The Black Caps had no need for despondency. Balls flew over or wide of the cordon with regularity. Trent Boult had Pope edging a ball short of Mitchell on 57, and tempted Root to clip a ball on 27 which was parried over the cordon by Tim Southee in the over before lunch.
Boult, coming over the wicket and angling across the incumbent right-handers, appeared the most potent member of the attack. He looks capable of replicating the unplayable delivery which had Zak Crawley caught for four in the second over of the innings.
In a potential setback for the Black Caps, wicketkeeper Tom Blundell was struck a blow on the right thumb from a Henry ball. He looked in significant discomfort wrenching off his glove, but the pain appeared to subside as the session progressed. His assured handiwork has been a tour highlight, in addition to his batting.