Jessop was offered a scholarship to play football at the University of Richmond; however, he declined the school's invitation and joined his favorite college team, the Virginia Tech Hokies, as a walk-on.[2]
He transferred from Virginia Tech to Shepherd University in 2015. As a backup quarterback, he was forced into action in the NCAA Division II semifinals when Shepherd's starting quarterback, Jeff Ziemba, was injured in the first quarter. Jessop led the Rams to a 34–32 victory over Grand Valley State.[3]
Jessop's first start for the Rams came on October 15, 2016, when he accounted for seven touchdowns (five passing, two rushing) in a 70–24 win over West Virginia State.[4] He set a school record in the game by passing for 480 yards, and the five touchdown passes tied a record; he was named the Mountain East Conference Offensive Player of the Week.[5]
Jessop became the Rams' full-time starter in the 2017 season, during which he threw for 40 touchdowns and just nine interceptions.[2] He was named the Mountain East Conference Offensive Player of the Week four times during the season.[6] At season's end, he was named the MEC Offensive Player of the Year, and he finished fourth in voting for the Harlon Hill Trophy.[7] He was named a Division II All-American.[8]
Jessop served as an assistant coach with Shepherd prior to the 2018 season.[9]
Though he was not selected in the NFL Draft, Jessop was invited by the Washington Redskins to attend their rookie mini-camp.[12] He was signed by the Redskins on August 26, 2018 after reserve quarterback Colt McCoy was injured, to serve as the backup quarterback during the team's final preseason game.[13][14] He was waived on September 1, 2018.[15]