Reese attended South Oak Cliff High School, where he played as a two-way tackle.
As a rookie, he was named the starter at left defensive tackle opposite John Meyers.
On July 12, 1964, he was traded to the Baltimore Colts in exchange for defensive tackle Jim Colvin.
[1] In 1964, although he was limited with an ankle injury early in the season, he passed Billy Ray Smith Sr. on the depth chart.
He was named the starter at left defensive tackle in the franchise's inaugural season, but suffered a knee injury in the second game against the Philadelphia Eagles and was lost for the year.