I believe you are correct. SC Law Section 16-23-20 states the following:
"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...
(9) a person in a vehicle if the handgun is:
(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 weapon 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;"
This can be viewed here.
https://www.scstatehouse.gov/code/t16c023.php
All SC firearms laws (state level) can be viewed here
http://www.sled.sc.gov/SCStateGunLaws1.aspx?MenuID=CWP
If any local jurisdiction gives you crap that is contrary to state law they are in violation of
"SECTION 23-31-510. Regulation of ownership, transfer, or possession of firearm or ammunition; discharge on landowner's own property.
No governing body of any county, municipality, or other political subdivision in the State may enact or promulgate any regulation or ordinance that regulates or attempts to regulate:
(1) the transfer, ownership, possession, carrying, or transportation of firearms, ammunition, components of firearms, or any combination of these things;"
This fight has happened before. The town of Irmo currently has an illegal ordinance on the books prohibiting carrying a "pistol" with no exemption for CWP but it isn't "enforced". Columbia and Richland county passed ordinances banning bump stocks and Columbia is currently being sued over it. Hopefully Richland County will be next followed buy Irmo. Municipalities here are attacking the 2A at an alarming rate.
I recommend you read over these laws to be sure. Don't have it on your person, don't have it out in the open. Console, glove box, or trunk only. The container/luggage crap is a gray area and ill defined. SC carry laws are not great, trust me, I live here.