One Dollar bill in jar is not from the 1940s.
When Wendell and his crew see the man coming towards them with the race result (of a race Wendell won but was not given credit for), the man first has his arms down to his sides, then across his front holding the paper with the result, then with his arms down on his sides again.
When Wendell takes Mary to the racetrack in his taxi, during closeups of them it is pitch black outside, but on long shots of the taxi outside it is early evening.
In many of the in-car shots during the races you can hear that they are manual transmissions being double clutched, but the drivers never take their hands off the wheel to shift.
It is Easter when Wendell Scott is arrested. Well Wendell leaves his house with a case of moonshine in his taxi and is cornered in town by the Sheriff, as he is trying to escape he hits a trailer full of watermelons. It is too early in the year to be picking watermelons in that part of the country.
On Wendell's ride home from the Army, his bus crosses a small bridge with Armco barriers along the side. Since Wendell started racing in 1953 and Armco barriers were not invented (in England) until 1954, there is no way that a bridge near Danville, Virginia would have them at that point in time.
Several characters in the film have much longer (wavy or Afro) hairstyles and facial hairstyles that were suitable for the 1970's but not for post World War II.