Puckett took part in FIS races during his junior years and was used in the Nor-Am Cup . At the Junior World Championships in Geilo in 1991 , he won the gold medal in slalom. In the 1991/92 season he won the overall and giant slalom classification of the Nor-Am Cup. He won his first points in the Alpine Ski World Cup with 17th place in the slalom in Chamonix on January 30, 1994. Four weeks later, he finished 7th in slalom at the 1994 Winter Olympics .
In the following winters, Puckett could not assert himself in the World Cup. For four years he did not achieve a single countable World Cup result. Often he was eliminated or could not qualify for the second rounds. In contrast, he continued to achieve good results in Nor-Am races. In the 1997/98 season he won the overall and giant slalom classification of the Nor-Am Cup for the second time. It was not until the 1998/99 season that he regularly made the points in the World Cup. His best World Cup placement by far was fourth place in the classic Hahnenkamm combination of Kitzbühel on January 21, 2001 .
In the Nor-Am Cup, Puckett decided in the 2000/01 season with four wins in the slalom discipline. In January 2002 he started a World Cup race for the last time, after which he only competed in alpine FIS races sporadically until 2006. The reason for this was his turn to ski cross. In 2004, he took part in the Winter X Games in Aspen and surprisingly won the gold medal there. He repeated this success in 2007.
From the 2007/08 season, Puckett was involved in the Freestyle World Cup . His first World Cup race on January 12, 2008 in Les Contamines , he finished second. Another second place followed a month later in Deer Valley . In winter 2008/09 he was classified seven times among the top ten (with a third place as the best result), which resulted in 4th place in the ski cross discipline ranking. With another podium at the beginning of the 2009/10 season, Puckett qualified for the 2010 Winter Olympics , where he was one of the oldest members of the American team. In 2011 he refrained from further starts in the World Cup and only took part in the Winter X Games, where he finished third.