Visiting Professor Visa for More Than Three Months?

I am a professor with a Fellowship at Stellenbosch University. The Fellowship is for three months, but my partner and I plan to stay longer. The Consular webpage says I simply have to return to my home country (US!) to renew the visa, but that can't be right. The University isn't being particularly helpful.

Has anyone faced this before? Did you simply go to Namibia or Swaziland and re-enter the country to renew for another three months?

Thanks for your help!