Find the Neutral Safety Switch and see if you can adjust its position in relation to the shift linkage movement. There may also be worn bushings in parts of the original '55 shift linkage.
Exactly. And if nothing happens now when you turn the key in the ignition, the neutral safety switch may now be burned out. Attempting to operate the starter using a mis-aligned neutral safety switch will greatly affect the life of the switch contacts. It is essential to get the switch adjusted so that the contacts seat dead on when the gear shift lever is in neutral.
First, check the rubber grommet in the lever that operates the shift link to the transmission. If that grommet has been softened by oil or if it is damaged it needs to be replaced immediately so that a proper adjustment can be made of the neutral safety switch.
Another similar problem can be at the bottom of the gear shift link where it goes into the transmission shift lever. If that pin that goes through the hole is worn real bad, or if the hole in the shift lever has been elongated, there will be a lot of slop in the link which can effect proper position of the neutral safety switch. All those things can be repaired with some patience and a few new parts.
Again, the quality of the chinese made repro neutral safe3ty switches is very substandard. They look good on the outside, but the materials, ampere ratings of the contacts and method of assembly of contacts to arm falls far short of original equipment. These units absolutely must be adjusted properly in order to get maximum life out of them and even so, they can be expected to fail in a year or so if the car is driven daily, that's with a solid proper adjustment.