Clancy had already secured gold in the men's team pursuit on Friday going into the final stages of Sunday's Omnium event.
After winning a bronze medal in the gruelling Omnium, Clancy said that it was hard to perform in both events in such a short space of time.
Clancy told BBC Sport: "I came here for the team pursuit and I put all my eggs in that basket. Mentally it was so hard getting up the very next day to go into a full blown omnium.
"I was in the form of my life, the kilo, the pursuit, the flying lap, I had a PB by a mile in all of them; it's just a shame that in the bunch races I got torn apart by the specialists.
"At one point I thought I was on for the gold, but that just slipped away."
Clancy's bronze in the Omnium secured Team GB's fifth cycling medal of the Olympic Games.