Smith was born in Tallahassee, Florida on May 6, 1993, as one of four children to Michael and Loretta Smith. He attended James S. Rickards High School, where he played American football as a safety, along with baseball.
Smith was drafted by the Milwaukee Brewers in the 13th round of the 2011 Major League Baseball Draft out of high school. He did not sign with the Brewers, instead playing college baseball at Santa Fe College. In his only collegiate baseball season, Smith hit for a .380 batting average, 17 extra-base hits, 17 runs batted in, and 31 stolen bases. He was then drafted by the San Diego Padres in the fifth round of the 2012 Draft. He signed with the Padres for a $375,000 signing bonus and made his professional debut that season with the Arizona League Padres. He also played for the Eugene Emeralds that year, hitting .305 with two home runs and 17 stolen bases between the two teams.
Smith opened the 2013 season with the Fort Wayne TinCaps. He played in 110 games, hitting .262/.367/.340 with four home runs and 64 stolen bases. The stolen bases were one short of the TinCaps record. Smith started 2014 back with Fort Wayne. During the season, he was promoted to the Lake Elsinore Storm. He finished the year hitting .310 with five home runs and led all minor league players in stolen bases with 88.
On December 19, 2014, the Padres traded Smith, Dustin Peterson, Max Fried, and Jace Peterson to the Atlanta Braves in exchange for Justin Upton and Aaron Northcraft. Smith was invited to attend spring training with the Braves, despite not having any Double A experience at the time. He began the 2015 season with the Mississippi Braves, appearing in 57 games, hitting for a .340 batting average and a .418 on-base percentage, before being promoted to the Gwinnett Braves of the Class AAA International League in June. Smith finished the season at Class AAA, hitting .306, with a .706 on base plus slugging percentage for the year. He was named the Braves' Minor League Player of the Year for all levels and added to 40-man roster after the season.
Smith started the 2016 season with Gwinnett. The Braves promoted Smith to the major leagues on April 11, 2016. He debuted that night against the Washington Nationals, starting in center field. Smith went 1-for-3 in his first game, with a strikeout and a single against Max Scherzer. Smith's helmet lacerated his forehead on a stolen base attempt in the fourth inning, and he left the game. Two days later, Smith recorded his first stolen base against the same team. His first career RBI on April 17 helped the Braves sweep the Miami Marlins. Smith hit his first career home run off of New York Mets pitcher Matt Harvey on May 3. On June 20, Smith fractured his thumb after Mets pitcher Antonio Bastardo hit him with a fastball. The next day, Smith was placed on the disabled list and expected to miss eight to ten weeks of the season. After a July reevaluation, his return date was moved to September. Braves interim manager Brian Snitker later stated that Smith would likely miss the remainder of the season. However, Smith returned on September 17, with two weeks left of the regular season, to serve as a bench player. Smith spent the 2016 offseason with the Naranjeros de Hermosillo of the Liga Mexicana del Pacífico and was released in October. He then joined the Indios de Mayagüez of Liga de Béisbol Profesional Roberto Clemente. Smith strained his oblique while playing in Puerto Rico and left the team in December.
On January 11, 2017, the Braves traded Smith, along with Shae Simmons, to the Seattle Mariners in exchange for minor-league pitchers Luiz Gohara and Thomas Burrows. That same day, the Mariners traded Smith, Ryan Yarbrough, and Carlos Vargas to the Tampa Bay Rays for Drew Smyly. As a Brave, Smith wore uniform number 17, but switched to 0 upon joining the Rays because his preferred number 13 was unavailable. A longtime fan of Friday the 13th, Smith said after his trade to the Rays that the franchise's main character Jason Voorhees represents his effort to be opposing teams' "worst nightmare." Smith has stated of his switch to zero, "My name's kinda different, my game is a little bit different, you know what, I'm gonna go with zero." On April 9, after reaching base five times (a single, double and three walks), stealing two bases, and scoring the walk-off run in a 3-2 victory against the Toronto Blue Jays, Chris Archer coined the term "Mallex Effect" to describe Smith's ability to wreak havoc and change games with his scrappy and speedy play. On August 18, Smith was optioned down to Triple-A Durham to make room for Kevin Kiermaier, who was returning off the disabled list. At the time of his demotion, Smith was hitting .279 with 16 stolen bases through 67 games.