The court ruling never had anything to do with use of USC. SC and Southern Cal had agreed back in the 1980s to both use USC.
SC was trying to trademark SC because people typically associate SC with South Carolina. All the court ruling did is prevent SC from trademarking SC because numerous other colleges also use SC such as Santa Clara. The court said that there is no evidence that people associate SC exclusively with South Carolina.
From SC's website,
Welcome to Trademark and Licensing Program
SC's trademarks include the word USC but not UofSC. In addition to those words, Logos and trademarks include: Interlocking USC, Interlocking SC (baseball/softball)
SO I misunderstood the court ruling and SC can still use an interlocking SC baseball hat logo. I saw some SC baseball hats available with the interlocking SC logo on an sports apparel website. It seems unlikely that SC and southern Cal apparel are both going to be available for sale at local retailers anyway given the distance between the two fanbases.
SC's decision to use UofSC does not appear to be based on this court ruling and I only see it used for social media purposes. They mostly likely use it for social media because southern Cal had already created USC handles on social media. UofSC isn't on any apparel that I have seen. SC probably doesn't have a problem using UofSC for social media given it tried to trademark SC.
SC's main logo is a C with a gamecock inside of it.