I would think you'd still be better off just staying in the house, especially if you can use your garage as a decontamination zone / airlock to avoid bringing a bunch of alpha emitters into the house.

Edit: assuming you're talking about fallout protection rather than the initial gamma burst. For that, yeah, get as many feet of earth between you and the source as possible; a trench will do that even without a vehicle over top.

If there's substantial fallout around you, being indoors won't save you. You need substantial shielding or distance between it and you for several days until it decays enough to not pose an immediate threat.

The "distance" part is interesting. Being in the middle of a tall building can work, because you're far enough from both the ground and the roof.

In much of TX you won't be able to dig a three to four foot deep trench due to limestone. It's one of the reasons we don't have basements.

Are there other reasons why basement-prevalence is so regional?

The water table. In some areas (like where I lived in Florida) your basement would flood. You'd have to have a pump working constantly.