West End Redbacks opener Jake Weatherald was denied a rare double ton to begin Day Two on Tuesday and Jake Lehmann (1) and Callum Ferguson (20) were also undone in swing-friendly conditions in the morning.
But it did little to slow the hosts from storming to 6/490 and a lunchtime declaration.
Wicketkeeper-batsman Harry Nielsen continued his strong form with the bat to craft an unbeaten 64*, which featured four boundaries and two sixes, as he shared an entertaining 62-run partnership with Tom Cooper (32 from 39).
Young paceman Wes Agar was rewarded with figures of 1/10 from 12 overs, including nine maidens, as the Redbacks went to work with the ball in the second session.
Two-time Neil Dansie Medallist Joe Mennie also claimed Tasmania’s George Bailey (1) cheaply in his first Shield outing of the season to put South Australia in the box seat.
But Tigers captain Matthew Wade (53*) and opener Alex Doolan (87*) remained steadfast in the third session, building an unconquered 107-run partnership to have Tasmania (2/176) at the close of play on Tuesday.