In 1936, at age 18, Tucker signed as a professional with the Fayetteville Bears of the Arkansas–Missouri League, but left the team after only two weeks, without playing a game.

[3] The following year, he was to play for the El Dorado Lions of the Cotton States League, but a back injury caused him to miss nearly the entire season.

[3] At the end of the season, Chicago White Sox farm manager Billy Webb was impressed enough to purchase Tucker's contract from Oklahoma City.

[5] At the beginning of the 1941 season, Tucker failed to win the final outfield spot on the White Sox roster from Dave Short,[6] and consequently spent 1941 at Oklahoma City, where he was coached by Rogers Hornsby.

[9] Of the Chicago White Sox outfielders at the beginning of the 1943 season, only Wally Moses was assured of a place on the team.

During spring training, manager Jimmy Dykes was impressed by Tucker, who was competing for the starting center fielder position alongside Moose Solters.

[12] Partway through the season, Tucker's performance caught the eye of American League President Will Harridge, who noted him as someone the public came out to watch in the absence of stars serving in World War II.

[13] Among Tucker's achievements during the year were a walk-off home run on July 26 to win a game against the New York Yankees 2–1.

[16] His hitting and fielding abilities impressed critics: sportswriter Fred Lieb noted him as a breakout performer that year,[17] and manager Jimmy Dykes called Tucker the finest defensive outfielder in the American League.

[21] Tucker's form faded after the All-Star Game; in early July, he had a hitless streak of 28 at-bats, causing his batting average to shrink from .375 to .327, resulting in losing his status as league leader.

[22] When his average fell to .320 after recording one base hit in 35 at-bats, he was removed from the starting lineup for a weekend matchup against the Detroit Tigers in an attempt to halt his decline.

[23] Tucker returned to the starting lineup shortly after being removed, and finished the season with a batting average of .287 and six triples.

[25] When the players returned to their teams at the end of the war, Tucker was slated to be the starting center fielder for the 1946 season, working alongside Wally Moses and Taffy Wright.

[26] Unlike the previous season, he struggled with the bat at first, and was relegated to his original status on the White Sox roster as a good fielder but a poor hitter.

[9] Tucker participated in game six of the 1948 World Series, scoring a run in the sixth inning on a walk en route to a 4–3 win for the Indians.

[9] Due to his hitting struggles, Indians manager Lou Boudreau tried converting Tucker to a switch hitter to start off the 1950 season.

[9] Tucker and Allie Clark both attempted to make the Indians roster to begin the 1951 season, as the additions of Harry Simpson and Minnie Miñoso made it likely that one or both of them would be traded or released.

[3] In the offseason, Tucker operated his own taxicab in Texas, and he contemplated retirement from baseball during a contract dispute in February 1952.