A quick run through 75 years of Chevrolet I find no listing for an Impala Custom CONVERTIBLE. The word “Custom” is only associated with the Impala Custom Hardtop which was essentially the Caprice Custom Hardtop in Impala trim. Generally, a vehicle labeled “Custom” was not available with a six-cylinder engine.
Therefore, it appears that the label “Custom”, on your convertible has been added later in the life of the car. Probably borrowed from an Impala Custom Hardtop.

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Chevradioman http://www.vccacolumbiariverregion.org/If I had known that growing old would be this much fun---I'd have done it sooner!