The flagging home side will need to chase an enormous score in the fourth innings after England went to stumps at 378-5 in Wellington on day two.

Captain Ben Stokes resisted any urge to declare, instead cementing England's position of power as four batters posted half-centuries to build on their first-innings advantage of 155.

Joe Root was at the crease on 73 while Stokes struck a whirlwind 35 not out against a tiring attack, after Jacob Bethell and Ben Duckett were both dismissed in the 90s.

The fast-moving nature of the Test means the Black Caps should theoretically have ample time to reach any target, as they bid to square the three-match series after losing the opener in Christchurch by eight wickets.

However, history is firmly against them. The highest successful fourth-innings chase at the Basin Reserve is 274, achieved by Pakistan against the hosts in 2003.

New Zealand's hopes of getting back into the Test were scuppered in the opening 40 minutes of play when they lost their last five wickets to be all out for 125.

Atkinson (4-31) removed the last three wickets with successive deliveries to become the first Englishman to claim a Test hat-trick since Moeen Ali against South Africa seven years ago.

The 26-year-old seamer was all smiles after bowling Nathan Smith for 14, then having Matt Henry caught in the gully and trapping Tim Southee lbw.

"It was great. As a bowler you think about getting five-fors and 10-fors and that sort of thing," Atkinson said.

"To get a hat-trick, I wasn't thinking about it too much at all but there's not many chances to do it and I was feeling pretty good when I was running in for that third ball.

"The plan this morning was to go hard at them. It worked out pretty well and the boys did really well with the bat, and we've got a pretty healthy lead."

Atkinson said he wasn't aware of Stokes' possible declaration plan but said he would be ready to go when called upon.

England made batting look comparatively easy, most notably when Bethell (96) and Duckett (92) combined for an untroubled 187-run second-wicket stand.

Bethell fell agonisingly short of a maiden Test ton when edging Southee to wicketkeeper Tom Blundell.

The 21-year-old looked crestfallen as he exited after a 118-ball knock which featured 10 fours and three sixes.

Atkinson said he felt for the Test newcomer.

"It's a shame, he batted so beautifully. It would have been nice for him to get his first hundred but he'll take a lot from that. He was outstanding."

Opener Duckett was closing on his fifth Test ton when he played-on off Southee (2-72), ending an innings of 112 balls.

First-Test centurion Harry Brook reached 55 before being caught in the deep off spinner Glenn Phillips while Root put some modest recent form behind him to post another batting milestone.

Root became only the fourth player to reach 50 runs in 100 different Test innings.

Ollie Pope fell for 10 off seamer Henry (2-76), who earlier dismissed Zak Crawley for eight, continuing a lean series for the opener.

In 19 Test innings against the Black Caps, Crawley has scored just 193 runs at an average of 10.15.

Henry said New Zealand needed to remain positive.

"Obviously we'll have a wee reflection overnight and we've got a big job to do tomorrow," he said.

"I think the boys will look forward to getting stuck in and getting into the fight. 

"There's still plenty of time in this game."