You're a rare fellow, being a Unionist from South Carolina. While anti-secessionist sentiments were not uncommon in other states, SC was one of the least divided states on this issue. The government and political culture of South Carolina had been dominated by coastal planters who managed giant plantations since early colonial times. When the SC secession delegates met in December 1860, the vote to secede was unanimous. Compare this to January 1861's secession convention in Mississippi (only 1 month later, no one has died in battle yet) where 15 Mississippi delegates voted against seceding. Still a minority, but indicative of unionist sentiment in another Southern state whose economy was totally dependent on slavery. Pro-Union sentiment led to guerilla movements in different regions of the Confederacy (Jones County, Mississippi, Scott County, Tennessee), I'm not aware of any such action by white South Carolinians.

Union men in almost every southern state formed volunteer regiments in the Union army. Every state, that is, except South Carolina. These were usually formed in nearby Union states or Union-controlled areas by exiles (i.e. the 1st Alabama Cavalry was formed in Union-occupied Memphis by Alabama unionists). Potentially you could have run away to another state and joined a non-SC Union volunteer regiment, but there would be no home-grown regiment of SC exiles for you to feel at home in.

So you're feeling very lonely in early 1861. Your best option might be passive resistance, such as not joining the CSA army. Such avoidance might have lasted, despite social disapproval, until the Confederate conscription acts passed in 1862 started drafting men into the army. You might have gotten away for a while if you were too old or too young, until the draft was expanded in 1864 to all white men aged 17-50 for the duration. Exemption to the draft could be found if you were rich enough to have 20 or more slaves, but if you were a wealthy slaveholder you probably weren't pro-Union.

If you ended up drafted but you still wanted to get away from the Confederacy, you could always desert, though the penalty for desertion was death. Not something to take lightly, but after mounting losses from 1863 onward, more and more men deserted the Confederate Army, many of them returning to their home communities. There was then the risk of being captured by the Confederate cavalry which rounded up deserters and forced them into service in the late stages of the war, but maybe if you could evade capture until the end of 1864, you could make it to safety across the Union lines in neighboring Georgia where Sherman's troops reached Savannah in 1865.

So, Mr. Exceedingly-Rare South Carolina Unionist, your best options are to avoid military service as long as you can, desert, and then hide.

References: For the planter domination of SC's political culture: Rachel N. Klein, Unification of a Slave State: The Rise of the Planter Class in the South Carolina Backcountry, 1760-1808 (Chapel Hill: University of North Carolina Press, 1990).

For discussions of Seccession conventions: Timothy B. Smith, The Mississippi Secession Convention: Delegates and Deliberations in Politics and War, 1861-1865

For Unionist guerilla activity and the motivations behind it, as well as Confederate treatment of deserters: Bynum, Victoria. (2003) The Free State of Jones: Mississippi’s Longest Civil War. University of North Carolina Press.