Smith batted on after he was hit by Anrich Norje in Durban, before staying off the field on day four once Australia declared.
The 37-year-old then passed checks for delayed concussions that evening and the next morning, before returning to the match on day five.
Smith has experienced concussions before, most notably when he was hit by Jofra Archer during the 2019 Ashes and missed the next Test.
"I was a little bit anxious to be honest after the hit," Smith said on Kayo Sports ahead of the start to the second Test.
"I've had a few in the past, and had a few concussions and I really don't like the way that makes me feel.
"The rest of that day I wasn't travelling too well. I think it was more to do with anxiety of hoping not to feel the way that I felt in the past, waking up the next morning and things like that.
"Fortunately I woke up well. I passed all the tests and things recovered pretty quickly."
Smith made scores of 25 and 27 in Australia's 214-run win in Durban, but his bigger contribution was his brilliant catching in the slips.
The right-hander's future has long been a talking point, with Smith often teasing that he could decide at any point he is ready to retire.
But Australia's captain Pat Cummins insisted in Gqeberha that may in fact be a positive.
"A lot of people have a grand exit and plan it years in ahead. But where he's at, and I totally get it, is 'I don't really want to think that far ahead'," Cummins said.
"That doesn't necessarily mean that something's imminent. I think it means he's in a really good place.
"You don't need to read into it anymore than that ... He's batting really well, catching really well. He seems to be enjoying himself around the group.
"Hopefully he's playing for a lot longer, but any conversation that I have isn't for a big grand exit that's very well planned ahead."
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