Three wickets fell in the middle session, two of them two to Miguel Cummins, including Kane Williamson for 54, in a venomous onslaught.
But the West Indies did not help their cause with sloppy fielding and dropped catches by Shane Dowrich and Shai Hope.
At tea, New Zealand were 190 for five in their second innings with Taylor on 56 and Colin de Grandhomme on 12.
The West Indies highest total in three Test innings so far on tour was 319 in their second bat of the first Test which they lost by an innings.
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But between lunch and tea they lost three for 94 with Williamson the first to fall after adding only three to his score.
The New Zealand skipper's timing deserted him as he played too late to a Cummins yorker which crashed into the middle stump.
It gave right-arm quick Cummins a wicket with his first ball in both the first and second sessions of the day after he had Jeet Raval caught and bowled for four in the morning.
In a barrage of short deliveries, he had Mitchell Santner edging towards gully and through the hands of wicketkeeper Dowrich before clobbering the batsman on the head when Santer ducked into a ball that did not rise as high as expected.
But when the short-pitched salvo ended it was spinner Roston Chase who accounted for Santner, caught at long-on for 26.
Taylor, who was dropped by Hope in the gully on 35, has faced 127 balls for his 56.
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