The 35-year-old left-arm seamer Payne had played two matches for Sunrisers in the tournament this season but has been force to withdraw due to an ankle injury.
Coetzee has previous IPL experience having represented Mumbai Indians in 2024, taking 13 wickets in 10 matches and with Gujarat Titans last year where he took two wickets in four games.
The 25-year-old played for the Proteas in the five-match T20I series in New Zealand last month, picking up eight wickets to help the visitors to a 3-2 series victory.
Sunrisers have been ravaged by injuries this year, especially to fast bowlers, as England international Brydon Carse was replaced by Sri Lanka's Dilshan Madushanka.
Good news from the Sunrisers camp is that regular captain Pat Cummins has been cleared to join the team after undergoing scans on his back in Sydney.
Sunrisers have won two and lost three of their opening five games so far with their next fixture against Chennai Super Kings at home on Saturday.