Adcox’s Winston Cup career started in 1974, running a handful of races for father Herb Adcox with sponsorship backing from the family’s Chevrolet dealership in Chattanooga, Tennessee.

In that year’s Winston 500 – considered one of the crown jewels of the sport – held at the fastest track on the circuit, the Talladega Superspeedway, Adcox and his father qualified for the event. With a hundred laps in the books, a caution came out as Donnie Allison’s clutch burned out and David Sisco’s motor went up in smoke. Gary Bettenhausen, who had pitted a lap after the leaders, and was up on jacks as young Adcox came down for service. Adcox’s car hit an oil and water patch and slammed straight into the Bettenhausen Matador, crushing catch can man Don Miller between the cars. A young crew member of the Nord Krauskopf team who was pitted nearby, Buddy Parrott, came rushing down to help, while Penske crewmembers John Woodward and John Watson were also injured. Miller was taken to the hospital and eventually had his right leg amputated. Learning of the extent of Miller’s injuries, Adcox went into shock, and his car was withdrawn from the event.

Adcox continued to run sporadic Winston Cup races among the years. Adcox qualified for the 1975 Talladega 500, but his crew chief Gene Lovell suffered a heart attack and died in their garage. Adcox’s car was withdrawn but he was able to find a substitute ride. However, the race would be delayed for a week due to rain and Adcox was again forced to withdraw. His spot on the grid was given to Tiny Lund, who had not raced in Winston Cup in several years. Lund would lose his life during the race in an accident on Lap 7.

The 1978 Winston 500 at Talladega would be Adcox’s finest hour, as Krystal Restaurants came on board for a partial season with sponsorship money and in his family-owned Chevrolet. Adcox finished a strong fifth in his career best Winston Cup finish.  But if Adcox’s career had a theme, it was being strapped for cash. A year later, he was again sponsorless – he timed in fourth for the 1979 Firecracker 400 at Daytona International Speedway but his car’s raw speed did not pay off, and his motor expired early. He would continue to run sporadic events over the years but never again contended.

Adcox’s primary claim to fame was for being a strong competitor in ARCA competition, a lesser stock car series that utilizes older Winston Cup cars and more grass roots based operations. He was especially dominant on the superspeedways—he won a record four of them in 1986 and had eight total superspeedway triumphs in his career. Five victories coming at Talladega.  For 1989, Adcox had signed up with Bumper to Bumper All Pro Autoparts and in a family-owned Chevrolet planned to run in a handful of Winston Cup events.

In the season finale at Atlanta Motor Speedway, Adcox crashed heavily on lap 198 of the event and died of major chest and head injuries, also suffering a heart attack as result of the crash. Upon investigation, it was determined that the severe impact had torn his improperly mounted racing seat away from its mount entirely, and this led to Adcox’s death. It also led to new safety regulations on the way seats were mounted for the 1990 season.  ARCA’s sportsmanship award, the H.G. Adcox Award, is named in Adcox’s honor, and is awarded annually by his father Herb. The Grant Adcox Memorial is also held annually in his honor at North Georgia Speedway and a subsequent event has been held also in Cleveland, Tennessee.