If it is a 440-T4, there are no shift solenoids and if that is the case, check to see if you have a vacuum system that works properly. It could also be a stuck 1-2 shift valve. Trapped vacuum can cause a 2nd gear start, which was a common problem.
If it is a 4T60-E, a bad 1-2 solenoid will cause more than a 2nd gear start.
We need to figure out if your trans is a 440-T4 or a 4T60-E as that was about the time when they were transitioning transmissions.
Will it take off in 1st when you put the selector in 1?
Does it shift 2-3 and 3-4?
1957 Nash Metropolitan
1960 Bonneville Conv 3x2 1964 Bonneville Vista
1965 Bonneville Conv
1965 2+2 455 3x2 and 1966 421 2+2 3x2 (donor car)
1967 GP Conv