Puckett began his career with
Southampton during the 1970s and played nearly 100 games for them in the
First Division between 1978 and 1986, scoring 14 times. However, he struggled to hold down a first team place and was transferred to
Third Division side
Bournemouth in 1986, helping them win that division's title and promotion to the
Second Division a year later. During his time at Bournemouth he played on loan for
Stoke City and
Swansea City. In 2011 Puckett signed for
Wessex League club
Lymington Town and on 26 November 2011 scored to become the league's oldest ever goalscorer at the age of 51. He later broke this record while playing for
Totton & Eling when scoring at the age of 53 in February 2014. He made fifteen league appearances for Totton & Eling, scoring one goal. On 21 May 2014 Puckett played a 4th division game in Iceland for Hvíti Riddarinn (The White Knights). Puckett was substituted in to the game in the 66th minute along with his son Bill who is one of two coaches of the team. The Knights won the game 2–0. ==Coaching career==