Jenkins made his professional debut in 2010 for the Johnson City Cardinals. He appeared in two games, allowing no runs over three innings pitched with two strikeouts. He returned to Johnson City in 2011 and had a 3.86 earned run average (ERA) with 55 strikeouts in 11 starts. Jenkins was personally invited to spring training by Cardinals manager Mike Matheny in 2012. He pitched that season for the Quad Cities River Bandits. He started 19 games with a 5.14 ERA and 80 strikeouts. Jenkins started 2013 with the Peoria Chiefs and was promoted to the Palm Beach Cardinals during the season. His season came to an end in August after having surgery on his shoulder. He finished the year with a 4.70 ERA in 13 starts. Jenkins returned to Palm Beach to start the 2014 season. He didn't pitch his first game until June due to the injury from the previous year. In 13 starts he had a 3.28 ERA and 41 strikeouts in 74 innings. After the regular season, the Cardinals assigned Jenkins to the Arizona Fall League.