Resuming at 8-309, Tasmania added 20 runs for their last two wickets overs at Blundstone Arena.
Tasmania started their innings well with a 33-run opening stand but then lost 3-5 in six overs.
They were 3-51 at lunch with captain Jordan Silk on nine and Ben McDermott six.
Play was held up for a few minutes shortly before lunch when Silk needed treatment for a cut on his right hand after he faced a delivery from paceman Nathan McAndrew.
The normally fluent McDermott faced 26 balls for his runs.
Opener Caleb Jewell (21) was caught off Brendan Doggett at the second attempt at first slip by Jake Lehmann, after the ball was knocked up by second slip Ben Manenti.
Matthew Wade fell for a six-ball duck, caught behind by Jake Doran from a ball that cut back.
Doran, who was off the field for part of day one after testing positive for COVID, was back behind the stumps within 20 minutes of play starting on Friday, after former Australian Test captain Tim Payne filled in for part of Thursday.
Doggett (2-12) then trapped Tim Ward lbw for 14.
Earlier, SA's innings finished after McAndrew was caught at point by Wade for 29 and Ward took a brilliant leaping catch at gully to dismiss Wes Agar.
Bird finished with 3-68 off 24.2 overs while fellow veteran quick Peter Siddle, who didn't bowl on Friday, recorded the best figures of 4-59 off 25.
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