Malcolm Turnbull is aiming for “boring, predictable, competent government” according to one of his closest colleagues, as a new opinion poll shows the leadership change has shot the Coalition back to an even-pegging position with Labor.

The Turnbull-led Coalition has 50% of the two-party preferred vote, according to the ReachTel poll, taken for Channel Seven on Tuesday night. Last month the Coalition was stuck in the election-losing position of trailing Labor by 47% to 53% after preferences.

Turnbull was also preferred prime minister by a massive 62% of voters according to the poll, with Labor leader Bill Shorten preferred by only 38% in the poll of 3,278 voters.

Turnbull is taking his time to consult on key decisions and reach out to those who will be key to his legislative success.

He spent his first full day as prime minister calling all eight of the Senate crossbenchers, taking a congratulatory call from US President Barack Obama, signing an agreement with two state premiers and calling others. And he’s said his new ministry won’t be sworn in until Monday.

“What we want is boring, predictable, competent government,” said senator Arthur Sinodinos, former chief of staff to John Howard and close Turnbull supporter.

“He’s been leader once before and there were fireworks and he’s had six years to reflect on that. When you get a second break like this you really want to hold on,” Sinodinos told Sky News.

Labor continued to put Turnbull under pressure over his promise to retain Tony Abbott’s Direct Action climate policy, given his previous strident attacks on it and insistence it could not meet deeper greenhouse gas reduction targets in an affordable way.

Sinodinos said Turnbull would deploy “pragmatism in pursuit of principle ... we will be pragmatic if it’s on the road to something. Turnbull says the important thing is to achieve reductions in greenhouse emissions”.

Turnbull is also constructing a new prime ministerial office, and sources did not discount that former treasury secretary Martin Parkinson could be offered the position of chief of staff – although no offer has been made.

In between constructing his new frontbench and managing a parliamentary sitting week, Turnbull sought to build bridges with the crossbench senators who will determine the fate of his legislative agenda, telling them he wanted “good communication”.

Liberal Democrat senator David Leyonhjelm said it was a short “hello how are you” call, with an agreement to talk again.

“I’ve said publicly Malcolm Turnbull could help the government’s cause in the Senate. His predecessor never tried very hard. He left negotiations up to his ministers, and some of them weren’t all that good at it.”

Palmer United party senator Dio Wang said he thought the early conversation with Turnbull was a good sign.

Senator Glenn Lazarus told Radio National on Wednesday he also expected Turnbull to be easier to work with.

“I was pretty frustrated with Tony Abbott, I’ve got to admit with my crossbench colleagues. They had the same opinion, there was no enthusiasm to meet with us. It was very difficult to get a meeting with him and in the 15 months or so I have only had one meeting,” whereas Turnbull “seems to be prepared to listen, and that was something Tony Abbott didn’t do”.

Turnbull is also considering his new frontbench, which will be sworn in Monday. Social services minister Scott Morrison is widely tipped to take treasury. Joe Hockey said he had had “discussions” with the new prime minister, but declined to say whether he had been offered an alternative portfolio. The immigration minister, Peter Dutton, has offered his resignation by text, but said he was also prepared to stay and serve in the Turnbull ministry. Turnbull said he had not spoken to Dutton.

It has been speculated that the education minister, Christopher Pyne, could become defence minister, although some in the Coalition thought this would be a mistake given the conflict between objective decision-making and his self-interest as an MP with a slim margin in South Australia, where voters are demanding Australia’s future submarine fleet be built.

Turnbull said he would definitely like to include more than the two women in the cabinet.

“There is no greater enthusiast than me for seeing more women in positions of power and influence in parliament, in ministries right across the country,” he said.

“I can assure you that. I am very committed to that, but I am not going to say any more about the new ministerial arrangements. You don’t have long to wait.”

Turnbull attended a prearranged signing for the national disability insurance scheme with NSW Premier Mike Baird and Victorian Premier Daniel Andrews and also called Queensland Premier Annastacia Palaszczuk.

Meanwhile, ministers, unsure whether they will still have their jobs by the end of the week, went about their ministerial business.

Hockey announced legislation to crack down on multinational tax evasion, the defence minister, Kevin Andrews, announced the first successful bombing raids on Syria and the trade minister, Andrew Robb, introduced enabling legislation for the China-Australia free trade agreement.