Smith, serving a 12-month ban from international cricket for his part in the Cape Town ball-tampering scandal, missed the Tridents' defeat to the St Lucia Stars and would play no further part in the Twenty20 tournament, his team said.
"He had a slight tear and he had to go back, so it’s a really unfortunate situation," Tridents captain Robin Singh said.
The injury is not expected to prevent Smith returning to domestic cricket later this month.
Although banned from playing for his home state New South Wales, Smith is scheduled to turn out for his Sydney club side Sutherland on Sept. 22, Cricket Australia's website (cricket.com.au) said.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
Signed as a replacement for Bangladesh's Shakib Al Hasan, Smith scored 185 runs in seven innings for the Tridents and claimed three wickets.
His ban ends in March 2019 but he will be suspended from leadership roles for a further 12 months.
(Reporting by Ian Ransom; Editing by Peter Rutherford)
