Moore signed with the Rangers for a $10,000 bonus and debuted professionally in the summer of 2015 with the Low–A Spokane Indians, moving up to the Single–A Hickory Crawdads for four games in September.

After the regular season, Moore played for the Salt River Rafters of the Arizona Fall League.

Moore spent 2017 with the Double-A Mississippi Braves, producing a batting line of .207/.291/.292 with 7 home runs and 42 RBI in 122 games.

[5] Moore started 2018 by returning to Double-A, this time with the Biloxi Shuckers, before being promoted to the Triple-A Colorado Springs Sky Sox in May.

[5] He hit a combined .299/.363/.522 with 14 home runs and 58 RBI in 121 games and was named a mid-season All-Star with Colorado Springs.

[12] He was on the Mariners 2019 Opening Day roster,[13] making his MLB debut on March 20 in the Tokyo Dome as a defensive replacement at third base in the seventh inning.

[15] Moore's season ended early, as he was put on the concussion list on September 22 after being hit in the head by a pitch from Brandon Bielek of the Houston Astros.

[16] On July 26, 2021, Moore hit a 395-foot grand slam in the bottom of the eighth inning to give the Mariners an 11–8 lead over the Astros,[17] which would be the game's final score.

[19] Following a solid 2020, 2021 was one of the worst offensive seasons of Moore's career, with a slash line of .181/.276/.334 with 12 home runs, 43 RBIs, and 21 stolen bases in 126 games.

[15] Moore improved offensively in 2022, with a slash line of .224/.368/.385, good for an OPS+ of 122, with 6 home runs, 24 RBIs, and 21 stolen bases in 104 games.

[22] He turned things around, with a two-home run game against the Minnesota Twins on July 26,[23] and showed more power in the second half of the season.

[22] Moore got the most playing time of his career in 2024 and continued to combine defensive versatility with league-average offense dragged down by a low batting average.