Smith returned to Cape Town last week after steering the Proteas to an innings victory against the hosts at the Oval, arriving just in time to see his wife Morgan give birth to a daughter, Cadence.
The 31-year-old, whose place as skipper was filled by AB de Villiers for a two-day game at Worcester, was due to return in time for the start of the second match of the three-Test series, which begins at Yorkshire's ground, Headingley, on Thursday.
JP Duminy said the whole side was delighted for Smith and added: "It's always good to have him back, the leader of our team. We're looking forward to having him here again."
South Africa's innings and 12-run victory against England was their first win at the Oval and came after an unbeaten first-innings triple century from Hashim Amla plus centuries from veteran all-rounder Jacques Kallis and Smith himself.