Unless there is something unique in South Carolina law or there is am uncommon clause in the agreement, a name change would not render the non-compete invalid.
Just to add to the above answer, normally when there is a name change or a change in the corporate status of a company, there is also an agreement that all the rights, goods, etc of the old company are transferred to the new company. You shoul d, however, consult a qualified employment lawyer in your area to see if the non-compete is enforceable and valid. There are many issues with these types of agreements and sometimes they are contrary to public policy and can be avoided.