The Black Caps begin day three 131 runs behind South Africa's first innings total of 308.
In the period around when Taylor was forced to retire hurt with the score on 148 for 2, Morne Morkel, playing his first Test since January 2016, had returned for a second spell and opted for a burst of short deliveries.
"He's disappointed as we could all understand but the way he walked off it doesn't look too good".
After South Africa's early slump, Elgar set the tone for the revival with a 126-run stand for the fourth wicket with captain Faf du Plessis before being joined by Bavuma just before tea.
New Zealand wrapped up the South African innings five overs after lunch, with the last six wickets falling for only 56 runs.
Quinton de Kock (10), undoubtedly the world's best number-seven batsman, was uncharacteristically cautious and once again the strategy failed as wickets fell without many runs being added.
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SA are backing their pace attack to snap New Zealand's unbeaten run at Dunedin's University Oval when the first Test starts on Wednesday.
The pitch continued to play better and better and even though Tom Latham's torrid form continued at the top of the order (his worldwide scores this year read 7, 0, 0, 2, 0, 10), New Zealand found the going good and scored quickly on a surface that played true and offered little or no lateral assistance for the bowlers after Willaimson ordered the heavy roller to smooth it out between innings'.
Henry Nicholls (12) became the second victim of Maharaj after he was caught brilliantly in the slips by Hashim Amla but Williamson along with nightwatchman Patel (9 not out) ensured the Black Caps ended the day at 177 for three.
Dean Elgar and Temba Bavuma resumed with their fifth-wicket stand worth 81‚ and built it to 104 before a Neil Wagner bouncer hurried Elgar into a stroke he didn't have to play midway through the morning session. Batting time: 240 minutes. Still to bat: James Neesham, B.J. Watling, Mitchell Santner, Neil Wagner, Jeetan Patel, Trent Boult.
NZC confirmed Guptill would miss the rest of this season due to injuries to both hamstrings that ruled him out of the first three ODIs against South Africa. Umpires: Kumar Dharmasena, Sri Lanka, and Bruce Oxenford, Australia.