The visiting Tigers (364, 4-226) led the Blues (338) by 252 runs at close of play on Tuesday, but will likely need to make a sporting declaration early on day four to have any hope of victory.
Continue to bat and time will run out for either side to pursue a result.
"We haven't won a game this year so we've got to put ourselves in a position to potentially risk losing to try and win," said Tasmania paceman Gabe Bell.
"But I think there's enough in this wicket, as we've seen the last three days, to get 10 wickets tomorrow."
In the third session, Ward (79no) brought up a third half-century from as many first-class knocks this summer by hitting Australian white-ball spinner Adam Zampa to backward square leg for a single.
It came after the 26-year-old top-scored with 92 in the first innings to push the Tigers to their first-innings lead.
"Tim's batted really well this whole match," Bell said.
Beau Webster (33no) helped the Tigers beef up their total as he hopes to put himself in the frame as a dark horse for Test selection later this summer.
Despite the hefty loss to India in Perth, Australia coach Andrew McDonald has confirmed he plans to take the same XI to Adelaide.
But a spot could open up for Webster at some point following concerns over Mitch Marsh's workload, with the burly allrounder pulling up sore after Perth.
The Sheffield Shield's player of the year last summer, Webster looms as the allrounder next in line should Marsh need a rest.
Earlier, Webster showed off his athleticism with a low slips catch to dismiss Chris Green.
It was to be the first wicket for right-arm quick Bell (4-52), who went on to rip through the tail and ensure the Blues would be back in the sheds by lunch.
Matthew Gilkes (62) resumed at 38 before that, pulling Kieran Elliott's short ball for four to bring up his 14th first-class half century before chopping on for one of Matthew Kuhnemann's two scalps.
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