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The safest route would in fact be the Interstate highways - I-75 to Knoxville, then I-640/I-40 to Asheville, I-26 to East Flat Rock, then US-25 and US-276 to Greenville. This is 340 miles and can be done in less than 7 hours.

There are no toll roads in that area of the country, and the only alternates are somewhat slow-going mountain highways. If you are planning on making the drive in one day, stick to the Interstates.