Senators will get their first proper chance to interrogate officials about the budget and other concerns, including the online census debacle, in Senate estimates hearings this week.
Estimate hearings in May - usually a two-week session - were truncated to just two days in the same week the budget was delivered to fit Prime Minister Malcolm Turnbull's early election timetable.
Australian Bureau of Statistics officials will be quizzed on Wednesday by economics committee members about the census website being taken offline after a series of cyber attacks.
Labor is also keen to pursue government plans for Medicare, which the coalition insists is not facing outsourcing or privatisation as claimed by the opposition during the election campaign.
And it's likely to keep up pressure on the government over Attorney-General George Brandis' dealings with the solicitor-general, and the same-sex marriage plebiscite.
Debate on legislation for the doomed national vote will continue in the lower house.
Labor is concerned about the mental stress on the gay community from a divisive plebiscite.
Asked whether leaving the issue open for potentially years was equally damaging, deputy opposition leader Tanya Plibesek said it was very strong argument to have a vote in the parliament.
"We can settle this next week," she told Sky News.
MPs are also set to debate the plan to tax backpackers 19 cents from the first dollar they earn and one of the workplace relations bills used to trigger the double dissolution.
Both sides will be on the look out for procedural hazards after Labor caught the government out and minister Kelly O'Dwyer accidentally voted for an amendment criticising the coalition.
Cabinet secretary Arthur Sinodinos said senators like him had learned you had to be on guard all the time in parliament.
"I think in the House of Reps sometimes in the past, because they've had a majority outlook, there's not been quite the same psyche," he told reporters in Sydney.
"They are now getting a couple of good lessons in that psyche and I think things will be better from here."
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