New Zealand are contemplating the possibility of a rare Test win after following on, having dug in with the bat on Sunday to set up an absorbing last two days of the second Test against England.

The Black Caps unearthed some much-needed tenacity to reach 202-3 in their second innings at stumps on day three in Wellington, closing within 24 runs of the tourists.

Their gritty response after being asked to follow on was founded on half-centuries by openers Tom Latham and Devon Conway.

Kane Williamson was a patient 25 not out at the close, needing just four more runs to surpass Ross Taylor as New Zealand's greatest Test run-scorer.

The former skipper will resume on Monday alongside Henry Nicholls (18 not out) with New Zealand harbouring slim hopes of squaring the two-match series on a Basin Reserve pitch starting to offer turn and variable bounce.

Only England (twice) and India have achieved the feat of winning a Test after following on.

Latham says his team have belief they can topple the tourists if they can carry momentum into day four.

"Absolutely, you've always got to look at ways to win a Test match," he said.

"We certainly know there's a lot of work to do to reach that point but whether it's 200 or 250 (target), you never know what's enough.

"We've seen with the wicket, whether it's the old ball or the new ball, there's plenty of turn there.

"To be close to parity at this point is really pleasing."

A New Zealand victory looked a distant prospect when dismissed for 209 in their first innings midway through the morning session, 226 runs behind England's 435-8 declared.

Latham (83) and Conway (61) combined to frustrate England for 149 runs before the first wicket fell, unveiling combative qualities that had been absent in a series dominated by England's impressive seam bowling.

New Zealand's foothold in the game loosened when both openers and Will Young (8) fell in the space of 18 runs, all to spin, to leave them 167-3 before Williamson and Nicholls dug in.

"For me and Dev to set the tone, that was important," Latham said.

"We lost a couple of wickets but the work Henry and Kane did at the back end of the session was really important. Who knows what tomorrow will bring?"

England assistant coach Paul Collingwood defended captain Ben Stokes' decision to enforce the follow-on, when instead putting the Black Caps back out in the field could have shut them out of the contest.

"The mantra of this team is to take the aggressive approach," Collingwood said.

"There was no doubt in Stokesy's mind, if we got opportunity, that was how we were going to go about things.

"Like we've said before, we're not scared of losing and we want to get ourselves into position to win games."

Jack Leach was the pick of England's attack, taking 2-59 off 31 overs, with his ball to bowl Young a textbook delivery for a left-arm spinner.

England are still well placed to notch a seventh straight Test win and secure a 2-0 series sweep after their 267-run first Test triumph at Mount Maunganui.

The key will be removing Williamson.