Tom Blundell and Zak Foulkes have been called into the squad as replacements for Finn Allen and Adam Milne, who were both ruled out for the entirety of the series
Conway has also been ruled out of the second Test against Australia at home. Australia is leading the two-match series 1-0 after a 172-run win at Wellington. The left-hander was instrumental in CSK's fifth title win last year, scoring 672 runs in 16 matches.
New Zealand Cricket (NZC) on Saturday announced that scans following the opening Test against South Africa showed a new injury in the same location as the one on which Jamieson was operated last year.
New Zealand head coach Gary Stead opened up on picking Rachin Ravindra over Henry Nicholls for the forthcoming home Test series against South Africa and said that it was a "very tough decision".
The two-Test series between South Africa and New Zealand will take place from February 4 onwards till February 17, with two games set to be played in Mount Maunganui and Hamilton.