In your situation you actually have no "double-taxed" income.
Since you never actually (physically) worked in Illinois, your income from your remote job is not taxable by Illinois. If you had physically worked in Illinois while being a resident of SC, your income would have been taxable by both states. That's because SC can tax ALL the income of SC residents, regardless of its source. And Illinois can tax non-residents on income from worked actually performed in Illinois. If that were your situation, you'd pay taxes to both states, but you'd be able to take a credit on your SC return for the taxes paid to IL.
As @Hal_Al explained, since your employer should not have withheld IL taxes, you must submit a non-resident IL return, showing zero IL income, in order to get those taxes refunded. And the income from that job is 100% taxable by SC, your resident state at the time.