Record-breaker Kane Williamson stroked a superb century Monday to steer New Zealand to 483 all out in their second innings and set England a target of 258 runs to win the second Test in Wellington.
Former captain Williamson hit 132 and Tom Blundell was last man out for 90 as the hosts fought back after being asked to follow on at the Basin Reserve.
England were 48-1 in their second innings at stumps after opener Zak Crawley, on 24, was bowled by a delivery that nipped back from New Zealand skipper Tim Southee.
Ben Duckett was unbeaten on 23 and nightwatchman Ollie Robinson on one, leaving England needing 210 more runs on Tuesday's final day for a 2-0 series sweep.
The tourists won the first Test by 267 runs.
"We're excited about chasing that (target) down tomorrow," said England spinner Jack Leach, who took 5-157.
"We know how we want to go about it and it's a case of us sticking to that process. We're confident."
New Zealand need nine wickets to pull off only the fourth victory by a side following on in the history of Test cricket and square the two-match series.
"It would be pretty special," Williamson said of New Zealand's historic bid.
"It's pretty exciting going into day five and we want to make sure our bowling is on. There's all to play for."
The hosts resumed their second innings on Monday at 202-3, still trailing by 24, but Williamson led the charge with his 26th Test century.
It fell to England's new batting star and part-time seam bowler Harry Brook to claim Williamson's prized wicket.
After grinding down England's bowlers, Williamson was finally caught down the leg side by wicketkeeper Ben Foakes off a fine edge after a sixth-wicket partnership of 158 with Blundell.
Williamson trudged off to hearty applause from the Wellington crowd after dragging New Zealand back into the match.
It was his second standing ovation of the day, having eclipsed Ross Taylor in the morning session to become New Zealand's most prolific Test batsman with 7,787 runs.
His partnership with Blundell helped New Zealand fight back after England declared their first innings at 435-8 then scuttled New Zealand for 209, before inviting the hosts to bat again.
New Zealand can join an elite club if they win -- only England, twice, and India have won a Test after following on.
Williamson, who survived a close shout for a stumping when on 45, handed his side a fighting chance after frustrating England alongside Blundell, with support from Henry Nicholls and Daryl Mitchell before lunch.
Blundell hit Stuart Broad to the boundary as New Zealand passed 400, then Williamson also smacked England's seamer for four to bring up his hundred -- his first since scoring two tons against Pakistan in December 2022.
New Zealand folded quickly once Williamson departed and he was full of praise for Brook's bowling.
"He didn't miss his length, he was immaculate. He's a special player," Williamson said.
With the former captain gone, Michael Bracewell went in sloppy fashion, run out for six as he inexplicably failed to ground his bat.
Foakes whipped off the bails from Ben Stokes' laser throw in the deep when Bracewell should have been home comfortably.
New Zealand lost their last four wickets for just five runs, Blundell last out when he edged Leach to Joe Root at slip going for a big hit.