England lost two early wickets in reply to New Zealand's 375 as their hopes of saving the series faded after another Black Caps defining partnership involving BJ Watling in Hamilton on Saturday.
When New Zealand had the wobbles, their first-Test hero partnered debutant Daryl Mitchell in a 124-run stand for the sixth-wicket to get them up to a respectable total.
England were then in immediate trouble with Dom Sibley and Joe Denly removed as they reached stumps at 39 for two.
With only three days remaining England appeared to have little chance of building a big enough total to win the Test and square the series
Sibley was out for four, not long after he ducked into a Tim Southee delivery and was hit on the helmet.
Joe Denly was also out for four, caught off Matt Henry while Rory Burns made it to stumps unbeaten on 24 after being dropped twice.
Joe Root, who has been out of form in the series, was not out six.
While Southee and Henry had England under pressure the tourists had struggled to exert the same dominance while in the field for nearly two full days.
They removed both overnight batsmen early on day two and then lost the initiative as Watling and Mitchell staged their repair mission before Stuart Broad removed both batsmen either side of the tea break.
Watling, whose double century set New Zealand up for a comprehensive victory in the first Test, was out for 55 on the last ball before the interval while Mitchell departed in the fourth over after play resumed.
New Zealand's last four wickets added a further 60 runs, with Broad finishing with figures of four for 73 and Chris Woakes three for 83.
All-rounder Mitchell, the son of former All Blacks coach and now England assistant rugby coach John Mitchell, was unruffled in his maiden Test performance.
He brought up his 50 pulling Ben Stokes through mid-wicket for four in a 159-ball innings which included eight boundaries and one six.
An even more cautious Watling, who reached his 18th half century with an elegant cut to the cover boundary off Sam Curran, faced 192 deliveries for his 55.
It was slow going by the pair, but from New Zealand's viewpoint their innings lasted almost two days which left England with only a remote chance of winning the Test to square the series.
England had their tails up at the start of the day when overnight centurion Tom Latham added only one boundary to his total before he misjudged a Broad delivery seaming back at him and was bowled for 105.
Broad stayed in the action to catch Henry Nicholls for 16 off Sam Curran, which brought the untested Mitchell to the middle to join Watling and bat New Zealand to a position of strength.