Yes, SC requires that non-residents report all income earned in SC. You'll have to file a non-resident SC return in addition to your home state KY return. KY can tax ALL your income, including what you earned in SC, but you'll be able to claim a credit on your KY return for the taxes paid to SC, so you won't be double-taxed.
How do I file a nonresident state return? (intuit.com)
In TurboTax, be sure to complete the non-resident SC return before you do the KY return, so that the program can calculate and apply the credit correctly.