To accommodate his foot structure, Dempsey wore a custom, flat-front kicking shoe that had no toe box.

How 'bout you try kickin' a 63 yard field goal to win it with 2 seconds left an' yer wearin' a square shoe, oh yeah, and no toes either.

"[7][8] Dempsey is most widely known for kicking a 63-yard field goal as time expired to give the Saints a 19–17 win over the Detroit Lions on November 8, 1970, at Tulane Stadium in New Orleans.

His record was tied three times—by Jason Elam with the Denver Broncos in 1998, Sebastian Janikowski with the Oakland Raiders in 2011, and David Akers with the San Francisco 49ers in 2012[13]—before it was broken on December 8, 2013, by Matt Prater, who hit a 64-yard field goal.

[12] On Sunday, September 26, 2021, Justin Tucker of the Baltimore Ravens broke that record with a 66-yard field goal as time expired to beat the Detroit Lions.

This walk-off kick eclipsed both Dempsey's and Graham Gano's 63-yarder in 2018 as the longest field goal to win a game on its final play.

[17][18] After retiring from professional football, Dempsey resided with his wife Carlene, who taught history at Kehoe-France, a private school in Metairie, Louisiana, a suburb of New Orleans.