The Rockies acquired minor league pitcher Jeff Criswell in the deal.
Smith made 15 appearances for the Rockies last season, going 0-1 with a 7.50 ERA. The 27-year-old Smith spent a chunk of the 2022 season at Albuquerque, where he had 12 saves for the Rockies' Triple-A affiliate.
Criswell worked his way through the Athletics' minor-league system last season. He went a combined 4-10 with a 4.03 ERA while pitching at High-A Lansing, Double-A Midland and Triple-A Las Vegas. He also had 119 strikeouts over 118 innings and walked 45.
The right-handed Criswell was a second-round pick by Oakland in 2020. He was 4-10 with a 4.07 over 29 career appearances for the University of Michigan.