Andy, I've never owned a closed car with the roof mount antenna, just my '39 CS and '47 Convertible so I can't address the specifics of any differences between the open car and closed car units. I'm looking at an old Bob Drake catalog but the photos/illustrations in it are about as helpful as a Ford Parts Manual...
If it's any consolation, the re-popped antenna (I have on the '47) has never worked properly. When in the lowered position(it's about a 1/4 inch out from the plastic antenna stop.