The West End Redbacks ended the penultimate day holding a lead of 311 runs at stumps after aggressively advancing to 2/166 from 27 overs in their second innings.
Opener Henry Hunt followed his first-innings 132 with 69 from 70 balls, veteran Callum Ferguson struck an unbeaten 72 from 74 deliveries while Tom Cooper is also not out on 17 from just six balls.
The display followed a sporting declaration from Tasmania just after Tea at 5/345 still trailing the hosts by 145 as the two teams vie for a result.
Tigers skipper Matthew Wade made 89 before part-time spinner Cooper had him caught at deep square-leg.
Wade and Alex Doolan's unbeaten 170 from 403 balls made up the bulk of Tasmania's total with Test captain Tim Paine (29*) also unbeaten.
Cooper was the surprise for South Australia with the ball, claiming three wickets in the second session of the day to finish with figures of 3/76. The visitors lost 3/56 in a 20-over span from Cooper’s hand.
Wes Agar continued his tidy form to, incredibly, concede just 19 runs from his 20 overs. He also took a wicket and bowled 12 maidens.
Joe Mennie was SA’s other wicket-taker with figures of 1/46 and six maidens in his return from injury.