The Greens are calling for Dean Winter to shake his "majority mentality" as the Labor leader and Liberals court support to lead Tasmania.
Jeremy Rockliff has been reinstalled as state premier after the governor granted his request to form a minority government.
But a planned motion expressing no confidence in the Liberals and confidence in Labor is set to be moved when state parliament returns on August 19.
Neither the Liberals with 14 seats nor Labor with 10 won the required 18 to control the lower house in their own right after a snap poll produced a hung parliament.
Mr Rockliff and Mr Winter have no formal agreements of support with the 11 minor party MPs and independents.
The Liberals pitched a multi-partisan budget panel on Friday as part of draft stability agreement talks with crossbenchers.
Immense backlash to the 2025/26 state budget was partially responsible for the original no-confidence motion against Mr Rockliff in early June.
Incoming treasurer Eric Abetz, who will be sworn in with the rest of Mr Rockliff's cabinet on Monday, said there was an "appetite" for budget repair among crossbenchers.
The former senator refused to "pontificate on budgets past" and wouldn't reveal how his approach would differ to Deputy Premier Guy Barnett, who has been removed as treasurer.
Mr Barnett's most recent budget forecast Tasmania heading for $13 billion worth of net debt by 2028.
Mr Abetz described debt as "intergenerational theft" and hailed the national budget being whipped into shape as the biggest achievements of the Howard/Costello government that he served in.
"That's the challenge here in Tasmania as well now, to get the budget back into shape," he told reporters.
His instinct was not to raise the tax burden of Tasmanians to balance the books, but Mr Abetz acknowledged the panel would be open to "all possibilities".
"I hope that my place in Tasmanian history is not cemented by being the shortest-lived treasurer in Tasmania," Mr Abetz said.
Mr Winter met with all five independent crossbenchers on Thursday to outline his vision for a stable and lasting parliament.
Shadow attorney-general Ella Haddad said Mr Winter shared Labor's draft collaboration framework and spruiked plans to shake-up oversight bodies.
Labor would work across the chamber, including with the Greens, to win support for the motion, Ms Haddad said.
The Clark MP argued the crossbench had a choice between an "energetic, ready, progressive Labor party" or a "tired 11-year-old Liberal government who offers more of the same".
Tasmanian Greens Leader Rosalie Woodruff said her party was yet to have a "proper conversation" with Mr Winter and had been "basically blanked".
She said the Labor leader remained stuck in a "majority mentality" despite the party's primary vote slumping to 25.9 per cent, 3.1 per cent lower than the 2024 state election.
"It doesn't give us a lot of confidence about how he'd operate as premier," Ms Woodruff said.
Labor needs the Greens' five lower-house MPs and another three crossbenchers to back the motion for it to pass.
The Greens have not set a deadline for the "proper conversations" to be had to earn their support.
Compromising on major issues does not mean compromising on values, independent MP Peter George said.
"Until now Liberal and Labor seem to think that promising to treat the crossbench better in parliament is just about all they need to do to win support. It is not," he said.
