Hi guys, I just got back from Mozambique for my second visa 'extension', and I can assure you it's possible by leaving the country and coming back. In my case, no questions asked - even though it was a computerized border. Though, truth be told, someone else I know got refused at the border.. he was told that he had to fly back to his country of origin before he could get a new visa (and this was at the same borderpost where I went for my first visa-run, just south of Gaborone).
I heard from other travelers along the way that they got new visa coming from Lesotho, but also from others that that is not customary for the Lesotho border. So, I guess going to Botswana or Mozambique is your best guess, though still a bit of a gamble.
One word of warning though.. traveling to nicer climates makes the SA cold pretty much unbearable (especially here in Joburg
Worries go down better with soup than without.