I have read that every Confederate state except SC contributed at least one full regiment to the Union Army. I'm sure some individual S. Carolinians also remained loyal, but not enough to form a regiment. Although I'm sure they could have formed at least one colored regiment from SC.
Most loyal Americans from SC fought as members of other southern state's Union regiments. Mostly because they were forced to flee their homes or risk getting strung up for their Old Flag beliefs before the war even started.
There was also a regiment of former South Carolina slaves who fought for the Union. The 1st South Carolina Volunteer Infantry was formed at Hilton Head in 1863 and saw some action in SC and Florida around Jacksonville. It eventually became the 33rd Regiment US Colored Troops and was disbanded in 1866. They were commanded by Colonel Thomas Wentworth Higginson an abolitionist from Massachusetts.