Fast bowler Damien Wright and off-spinner Xavier Doherty claimed two wickets each as South Australia buckled to 128 for six in reply to the home side's first innings of 375 on Wednesday.
Doherty took 2-19 from 10 overs to add to the 35 runs he scored in a record-breaking ninth-wicket partnership with Michael DiVenuto earlier in the day.
Wright finished with 2-42 and Geeves 1-22, while Greg Blewett top scored for SA with 39.
Tasmania earlier milked 135 runs from their last three wickets, including a 119-run stand between DiVenuto and Doherty.
DiVenuto finished nine short of his 12th century in Sheffield Shield cricket while paceman Shaun Tait finished with 4-111.
In Hobart (day two): Tasmania 375 (M. Bevan 106, M. DiVenuto 91, D. Dawson 73; S. Tait 4-111, P. Rofe 3-50) vs South Australia 128 for 6 (G. Blewett 39; X. Doherty 2-19, D. Wright 2-42).