A brutal spell from Neil Wagner as New Zealand cruised to a 240-run victory at Seddon Park on Tuesday.
West Indies face record target 444 as Taylor posts New Zealand best 17th ton. They were 30-2 at stumps on day 3.
Tim and Trent adventures included one key partnership, four wickets and two spectacular catches, which left West Indies trailing by 158 runs at stumps.
West Indies late charge unravels New Zealand. At stumps, the host were 286 for seven.
West Indies are up against it to level the series, with the hosts high on confidence Southee set to return.
Marsh, Stoneman, de Grandhomme, Blundell and Brathwaite others to move up.