16-23-20. Unlawful Carrying of Handgun; Exceptions. It is unlawful for anyone to carry about the person any handgun, whether concealed or not, except as follows, unless otherwise specifically prohibited by law:
(a) secured in a closed glove compartment, closed console, closed trunk, or in a closed container secured by an integral fastener and transported in the luggage compartment of the vehicle; however, this item is not violated if the glove compartment, console, or trunk is opened in the presence of a law enforcement officer for the sole purpose of retrieving a driver’s license, registration, or proof of insurance; If the person has been issued a concealed weapons permit pursuant to Article 4, Chapter 31, Title 23, then the person also may secure his weapon under a seat in a vehicle, or in any open or closed storage compartment within the vehicle’s passenger compartment;
The way I read this, as long as you've got a CWP that's honored, it doesn't matter whether the handgun is a concealed weapon or not. The rules for centerfire rifles/shotguns are different.