The Kansas City Royals selected Puckett in the second round, with the 67th overall selection, of the 2016 MLB draft.[9] He signed with the Royals, receiving a $1.2 signing bonus. After signing, he was assigned to the AZL Royals, and after pitching to a 3.86 ERA in two games, he was promoted to the Lexington Legends where he posted a 2–3 record and 3.66 ERA in 11 games. He began the 2017 season with the Wilmington Blue Rocks.[10]
Chicago White Sox
The Royals traded Puckett and Andre Davis to the Chicago White Sox for Melky Cabrera on July 30, 2017.[11] The White Sox assigned him to the Winston-Salem Dash. In 25 total games started between Wilmington and Winston-Salem, Puckett posted a 10–7 record with a 3.98 ERA.[12] He missed the entire 2018 season with elbow pain, and elected to undergo Tommy John surgery on March 13, 2019.[13]
Puckett did not play in a game in 2020 due to the cancellation of the minor league season because of the COVID-19 pandemic.[14]
Atlanta Braves
On December 10, 2020, the Atlanta Braves selected Puckett in the minor league phase of the Rule 5 draft.[15] Puckett split the 2021 season between the High-A Rome Braves and the Double-A Mississippi Braves, accumulating a 4–6 record and 2.90 ERA with 72 strikeouts in 80+2⁄3 innings pitched across 20 games.
In 2022, Puckett returned to Mississippi. In 33 games, he struggled to a 2–5 record and 7.36 ERA with 74 strikeouts in 58+2⁄3 innings of work. He was granted free agency on November 7, 2022.[16]
Seattle Mariners
On February 8, 2023, Puckett signed a minor league contract with the Seattle Mariners organization.[17] In 47 appearances split between the Double–A Arkansas Travelers and Triple–A Tacoma Rainiers, he accumulated a 6–4 record and 4.52 ERA with 54 strikeouts across 63+2⁄3 innings pitched. Puckett elected free agency following the season on November 6.[18]
Charros de Jalisco
On April 23, 2024, Puckett signed with the Charros de Jalisco of the Mexican League.[19] In 18 games 17.2 innings of relief he went 3-0 with a stellar 1.02 ERA with 17 strikeouts and 3 saves.
Personal life
Between his sophomore and junior years of high school, Puckett suffered a brain injury. Doctors put him in a medically induced coma for three days.[1]