The San Diego Padres selected Smith in the 11th round of the 2011 Major League Baseball draft; he signed for a signing bonus of $250,000. Pitching for the Class A AdvancedLake Elsinore Storm in 2012, Smith went 9–6 with a 3.85 earned run average and 137 strikeouts. He started the 2013 season with the Double-ASan Antonio Missions. He was named the Texas League Pitcher of the Week in April 2013, and the Padres Minor League Pitcher of the Month for April with a 1.38 ERA over five starts. Smith made his Major League debut Saturday, May 11, 2013, against the Tampa Bay Rays as he was called up from Double-A. In his Major League debut, Smith pitched a clean first inning, striking out his first two batters, but he did not record an out in the second, giving up six earned runs on five hits. He was sent down to the Tucson Padres of the Triple-APacific Coast League on May 22 after allowing 15 earned runs in three starts. He was recalled on June 8, but did not make an appearance before he was optioned back to Tucson on June 10. He came back to the Major League club on June 28 after posting a 2.54 ERA in five starts overall with Tucson. Smith joined the bullpen where he made three appearances before being sent down again on July 5. Smith made seven more starts for Tucson, striking out 10 in his final appearance, and then returned to the Padres' rotation with the September roster expansion. He made four more starts for the Padres in 2013, allowing 10 runs with 31 strikeouts.
The New York Mets selected Smith in the Rule 5 draft on December 14, 2017, and immediately traded him to the Kansas City Royals for cash considerations. After a strong spring training, Smith was chosen as a member of the bullpen for Opening Day. In 2018 with Kansas City he was 1–6 with a 6.92 ERA in 38 games, in which he pitched 78 innings. On November 26, 2018, Smith was designated for assignment.
Milwaukee Brewers
On January 11, 2019, Smith signed a minor league contract with the Milwaukee Brewers. He was assigned to the San Antonio Missions of the PCL to start the 2019 season, for whom he was 6–3 with a 2.33 ERA in 15 starts, in which he pitched innings and struck out 85 batters. He had his contract selected to the major leagues on May 5. With the Brewers, he was 0–1 with a 7.82 ERA in innings. On August 9, Smith was designated for assignment.
San Francisco Giants
On August 12, 2019, Smith was claimed off waivers by the San Francisco Giants. With the Sacramento River Cats of the PCL, he was 1–1 with a 4.20 ERA in 15 innings in which he struck out 18 batters. With the Giants, he was 0–0 with a 2.08 ERA in eight innings. Smith was designated for assignment by the Giants on February 10, 2020.
Oakland Athletics
On February 15, 2020, the Giants traded Smith to the Oakland Athletics in exchange for cash considerations. This trade was the first swap involving a major league player between the Bay Area rivals since December 4, 1990, when the Athletics acquired Ernest Riles from the Giants for Darren Lewis and minor league starting pitcher Pedro Pena.