Adcox's Winston Cup career began in 1974 running a handful of races for his father, Herb Adcox, with sponsorship from the family's Chevrolet dealership in Chattanooga, Tennessee. At the 1974 Winston 500 at Talladega Superspeedway, Adcox's car hit an oil and water patch during a pit stop and slammed into Gary Bettenhausen's Matador, crushing catch-can man Don Miller between the two cars. Miller was hospitalized and eventually lost his right leg to amputation. Learning of the injuries, Adcox went into shock and his car was withdrawn. Penske crew members John Woodward and John Watson were also injured, and a young Nord Krauskopf team crew member named Buddy Parrott rushed down to assist.
Adcox qualified for the 1975 Talladega 500, but crew chief Gene Lovell suffered a fatal heart attack in their garage. Adcox found a substitute ride, but rain delayed the race a week and forced another withdrawal; his grid spot was given to Tiny Lund, who had not raced in Winston Cup for several years and died in an accident on lap seven of that event.
His best Winston Cup result came at the 1978 Winston 500 at Talladega, where Krystal Restaurants came on board with sponsorship money for a partial season. Adcox finished fifth in his family-owned Chevrolet, the career-best Winston Cup result. At the 1979 Firecracker 400 at Daytona International Speedway, Adcox timed in fourth in qualifying, but his engine expired early. He continued to run sporadic events over the following years without again contending.
Adcox's primary competitive base was the ARCA Permatex SuperCar Series, a lesser stock car series that used older Winston Cup machinery. He was especially dominant on superspeedways, winning a record four superspeedway events in 1986 alone and accumulating eight superspeedway victories across his career, five of them at Talladega. Dale Earnhardt, speaking ahead of a 1989 race at North Carolina Motor Speedway, remarked on Adcox's ARCA success and said that with the monetary backing other teams enjoyed he might have become a success in NASCAR.
For 1989, Adcox signed with Bumper to Bumper All Pro Autoparts and planned to run a handful of Winston Cup events in a family-owned Chevrolet. At the season finale — the Atlanta Journal 500 at Atlanta Motor Speedway — Adcox crashed heavily on lap 202 of the event and died of major chest and head injuries, also suffering a heart attack as a result of the crash. Investigation determined that his improperly mounted racing seat had torn entirely away from its mount during the severe impact, and this was identified as the cause of death. The accident led directly to new safety regulations on seat mounting for the 1990 season.
ARCA established the H.G. Adcox Award for sportsmanship in his honor, presented annually by his father Herb until Herb's own death in 2015. The Grant Adcox Memorial race is held annually at North Georgia Speedway, with a subsequent event also held in Cleveland, Tennessee. Adcox is interred at Chattanooga Memorial Park in Chattanooga, Tennessee.
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