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The 24-year-old qualified third after two routines reduced the 16-strong field to eight finalists, backing up his first round 52.415 with a second-round 59.705.
He lifted again in the final, recording 60.675 to finish third.
China's Dong Dong, who finished second at the 2016 Rio Olympics, took second with 61.235 with gold going to Belarussion Ivan Litvinovich on 61.715.
It's New Zealand's first medal at Olympic level in gymnastics and trampolining, five years after Schmidt finished seventh at the Rio Games.