You are correct that SC does not honor NM's concealed carry permit.
However, in South Carolina, you may keep a loaded handgun in the closed glove box or center console of your vehicle. You may carry it in any closed compartment. That compartment must have a door that can be closed but does not have to be locked. You may not put it in an open compartment like the map pocket on the door or under the seat. If stopped for a traffic violation, there is no requirement to advise the officer that it is there. However, if you need to enter that compartment to get your documents you should so advise the officer.
You may store it loaded or not in the trunk. You may also take it into your hotel or rental property. Carrying into a bar or restaurant the serves alcohol for on premise consumption is a misdemeanor. Carrying into a publicly owned building or school is a felony.
Hope that helps.