Bitchin' car, indeed.
In deference to mymokke, and given the photos I've found of him building the car, I think it was based on a 1964 Falcon. Given that the '64 model was released in 1963, and Jeffries tells of building the car in a hurry, it's possible it was a customized '64 that did tour auto shown in the later half of 1963.
Here is a '63 Falcon dash, a '64, and Jeffries working on the Python (with dash visible):