After graduating from Bellaire High School in Houston, Texas, Moore joined the Tigers as an amateur free agent in 1957 when just eighteen years old.

He caught 12 innings of a 13-inning marathon against the California Angels in his Major League debut, and his first MLB hit was a thirteenth-inning single that moved the eventual winning run to third.

He then was hired as bullpen coach for the Seattle Pilots shortly after their one and only season in Major League Baseball (1969), and went with them when they moved to Milwaukee and became the Brewers starting in 1970.

He rejoined the Rangers in 1980, but his third stint with them lasted just one season when, at the end of year, manager Pat Corrales and his entire coaching staff were fired.

Moore, who went 163–190 (.462) in his 2+ years managing the A's, was replaced on an interim basis on June 25 by Jeff Newman, and in early July by Tony La Russa, who stayed in Oakland through 1995.

He stayed with the Reds through Piniella's departure in 1992, after which he left for his fourth stint in Texas (1993–1994, during the managerial tenure of Kevin Kennedy), followed by three seasons as bench coach for the Colorado Rockies on the staff of Don Baylor.

In 2000, Moore was appointed the first manager in the history of the Round Rock Express, the Double-A affiliate of the Houston Astros owned by a syndicate that included Baseball Hall of Fame pitcher Nolan Ryan.

[6] During five seasons as the bench coach for manager Ron Washington, the Rangers won two American League pennants, qualified for the playoffs three times, and tied for the second AL wild card in 2013.

After the Rangers' defeat at the hands of the Tampa Bay Rays in a one-game playoff on September 30, 2013, Moore, 74, was notified he would not return as the club's bench coach for 2014.