Moore's first-ever race in NASCAR was in the
Grand National Series on August 2, 1964. He started the race in a 1963
Ford, but dropped within three laps due to overheating problems. Moore's best season of his racing career was in 1965, where he managed to grab three top fives, seven top tens, and even a pole, driving
Louis Weathersbee's No. 45
Plymouth and competing in just fourteen races. Moore's best season in the point standings, however, was in 1968, where he finished 29th in those standings. Bud's final race in his relatively short career was five years after his second to last race, at the 1973
Southern 500 at the
Darlington Raceway. He only completed 174 of the 367 laps, retiring from the race due to engine problems in his 1971
Mercury. ==Later years==