The tourists were 18-2 after openers Ben Duckett and Zak Crawley were both removed in the six overs bowled before stumps on day three in Hamilton.
It left New Zealand, dismissed late in the day for 453, needing eight more wickets to secure a consolation victory after England easily won the first two Tests.
It may only be seven wickets required, with doubt over whether the injured Ben Stokes will bat.
The England captain is being assessed for a hamstring injury suffered while bowling which forced him to immediately leave the field.
Any prospect of England reaching what would be a world-record fourth-innings score were quickly quashed when Duckett charged down the pitch and played on for four.
It handed a first wicket of the match to seamer Tim Southee, playing his 107th and final Test before retiring.
Crawley was trapped lbw for five to complete a miserable series in which he fell to paceman Matt Henry in all six of his innings, totalling just 52 runs in the process.
Jacob Bethell was at the crease on nine and Joe Root was yet to score.
"We wanted to try to get more wickets and then set ourselves something that we were going to be realistic that we could smash down," England batting coach Marcus Trescothick said.
"We're still in with an opportunity but it's obviously pretty tough."
Masterful Williamson
Rain wiped out the first session at Seddon Park but it didn't prevent Williamson advancing from his overnight 50 to a 33rd Test ton, before departing for 156.
Williamson put on 105 for the third wicket with Rachin Ravindra (44) and departed after tea when top-edging a sweep off Shoaib Bashir (2-170).
New Zealand's greatest run-scorer struck 20 fours and one six as he took his tally of Test runs at Seddon Park to 1,614, the most by any batter at any New Zealand venue.
He has scored centuries in five successive Tests at the ground, where he boasts an average of 94.94 from 21 innings.
Williamson warned victory wouldn't necessarily come easily and hopes spinner Mitchell Santner could be a decisive figure.
"When they bowled some spin it sort of spun and bounced, but the wicket is still a good one," Williamson said.
"As the ball gets a bit softer and the wicket becomes a bit friendlier to bat on, there'll be some hard yards out there.
"It has to be a big push tomorrow."
Daryl Mitchell hit out near the end of New Zealand's innings, caught in the deep for 60 to hand part-time spinner Bethell a first Test wicket and, ultimately, England's best bowling figures for the innings of 3-72.
Santner holed out off Root on 49 to fall just short of back-to-back half-centuries for the match.
Southee scored two in his last Test knock, failing to add to his career tally of 98 Test sixes.
Williamson wants New Zealand to send Southee out a winner.
"He's just an outstanding servant of the game and a huge player for us over the years, those achievements are there for everybody to see," Williamson said.
"His leadership and how he carried himself throughout his Test career, it's been a big part of why this team's been in a pretty strong place for some time."
