Redbacks opener Henry Hunt was all class on Friday when he made an unbeaten 136 to guide the Redbacks to 3-329 by stumps in Hobart.
But long periods of rain meant play didn't resume on Saturday until after 3pm AEDT.
South Australia added 69 runs in the space of 12 overs before declaring at 6-398.
Tasmania went to stumps on day two in all sorts of trouble at 4-96, with Nivethan Radhakrishnan (43), Jake Weatherald (12), Jordan Silk (18) and Tim Ward (7) already sent packing.
With just two days remaining, South Australia have the chance to enforce the follow-on if they can rattle through Tasmania's middle and lower order quickly on Sunday.
Day one was dominated by the Redbacks as opener Hunt and Jason Sangha (151) combined for a 300-run partnership.
The lengthy rain delay on day two meant South Australia needed to up the ante when play finally resumed in the afternoon.
South Australia's Jake Fraser-McGurk was in an attacking mood against Tasmania. (Matt Turner/AAP PHOTOS)
Hunt was caught behind without adding to his overnight score, but Jake Fraser-McGurk (43 off 37) and Jake Lehmann (24no off 25 balls) batted like it was a one-dayer to add some handy quick runs before the declaration.
Tasmania moved to a solid 0-45 in reply before Weatherald chopped Liam Scott (2-19) on to his stumps.
Radhakrishnan was caught chasing a wide delivery, before Redbacks speedster Nathan McAndrew (2-28) struck two late blows to leave Tasmania four wickets down.
Silk was out edging behind in the final over of the day. Â
12°C