Schmidt, who won bronze at the Tokyo Games, finished out of the medals after a tumble during his routine at the Bercy Arena in Paris.
"I knew I needed to push hard to get onto that podium - those boys are incredible athletes and posted some big scores. So I went out there and pushed hard, and it didn't come off."
Schmidt went into the medal round in fourth place, but went off-line during a twist jump and fell off the trampoline.
He was unharmed, but looked disconsolate as he waited for the score; Schmidt's 19.5 points placed him eighth overall.
The medal winners were from AIN and China.
"It's obviously devastating. It's been ... such an amazing build-up. I'd been training really, really well and the body was feeling good. I'm just disappointed I couldn't deliver on the day."
This was Schmidt's third Olympics, and the Auckland-based athlete finished fourth at the Cottbus World Cup in Germany.
"I've got more in me; I know I could do better. It's super frustrating, but hey, that's sport. The margins are so small and yeah, I had to give it everything to get on the podium, and I came up short."
Earlier, Maddie Davidson - Schmidt's partner and the other member of New Zealand's trampoline team - put together a great showing in Paris, finishing seventh in the women's final. Team GB's Bryony Page won gold.
