Jacqui Lambie, the Tasmanian senator whose vote may be critical to the success of the government’s tax cut plan, says she will seek widespread briefings before declaring her hand on the Coalition’s $158bn package.

Labor is also holding out its support for the entire three-stage tax plan, insisting the government split the legislation so that the first tranche, which delivers tax cuts for low- and middle-income earners worth up to $1,080, can be passed immediately.

The Coalition, however, is claiming it has a mandate for the entire package, and is negotiating with the crossbench to try to secure a majority in the Senate against the opposition from Labor and the Greens.

Lambie told Guardian Australia ahead of her confirmed election that once she was formally declared elected, she would consult widely on the proposed legislation.

“I will need to speak to ministers, advisers, interested parties, government departments … which at the moment I do not have access to,” Lambie said.

“It would premature of me to discuss any of the bills at this point in time.”

The Coalition is on track to hold 35 seats in the new Senate when parliament returns in July, and needs 39 votes to secure the majority needed to pass legislation.

If it gains the support of conservative Cory Bernardi, the Coalition will still need to secure three more votes from the Senate crossbench, expected to comprise Lambie, and two senators from each of One Nation and Centre Alliance.

The Australian Electoral Commission formally declared the Tasmanian Senate result on Thursday, with Lambie returned after she was forced to resign over dual citizenship in 2017.

The firebrand senator’s party, badged the Jacqui Lambie Network, has secured almost 9% of Senate first-preference votes in Tasmania – just over 30,000 votes – polling more than Pauline Hanson’s One Nation (3.45%) and Clive Palmer’s United Australia party (2.64%) combined.

The former soldier was first elected to the Senate as a member of the Palmer United party in 2013.

Hanson has indicated her two senators will not support the government’s tax cut plan unless the Coalition agrees to a new coal-fired power station and the revival of the 1930s Bradfield water scheme.

Centre Alliance, meanwhile, is demanding the government put in place a domestic gas reservation policy and take action to ensure tax cuts are not absorbed by higher energy costs.