I have owned several '58/60 F100s since 1958, currently own 3 '59's an one '60.. For starters the fuel tank is mounted behind the seat, therefore there is nothing to cover up except the tank.
I have a '59 F100 V8 4x4, big window custom cab that has about every accessory listed in the book, padded dash, padded visors, two heaters, underseat and fresh air under dash, radio and safety package. There is no cover over the fuel tank.
If the custom cab F100's had a cover over the tank it would have been cardboard similar to the head panel..
As a general rule the Custom cab GM trucks of the '60's had a cardboard cover over the fuel tank...
It should be kept in mind that the trucks of the late '60's and hence, were work vehicles, with the advent of RV's the pickup being a family second vehicle and thereby the term SUV's was coined.