At stumps on Tuesday, the West End Redbacks trail by 223 runs, finishing 7-170 in their first innings.
Veteran pair Callum Ferguson (51) and Tom Cooper (33) sparked a brief revival after SA were reduced to 3-59, but both were dismissed along with reliable in-form wicketkeeper Alex Carey. The three were out for 3-15 in the final overs of the day.
It was a strong all-round bowling performance from Tasmania, headlined by Tom Rogers who took 3-45.
Left-arm quick Nick Winter starred with the ball for SA, securing his fourth five-wicket haul in just seven innings as the Tigers were all out for an imposing 393 just after lunch.