The Coalition is removing support and the non-government parties are combining to refer Labor's move to a weeks-long Senate committee process.

The Opposition, Greens and crossbenchers voted in the Senate on Wednesday morning to refer the legislation, which Labor had wanted passed in Parliament this week, to a Senate inquiry with a reporting date of May 7, the first sitting date after this week.

The Opposition, Greens and crossbenchers are critical of the government's handling and lack of consultation over the matter.

Manager of Government Business in the Senate Katy Gallagher said Labor was trying to fix a "broken" immigration system inherited from the Coalition.

"No more lectures from those opposite. No more out trying to wind up the scare campaigns. No more accusations of us not working in the national interest, because we see a clear example today of the opposition not acting in the national interest," Senator Gallagher told Parliament.

"I can't count the number of times I've heard those opposite lecture about needing to move quickly, to address loopholes, to make sure that we're keeping the community safe, to putting us in the strongest legal position. And that ends today. You have no credibility on that anymore."

The government legislation has been hurried by a recent High Court decision, although the Home Affairs department insists the latest amendments are not driven by the prospect of another related High Court ruling due next month.

The amendments to the Migration Act would give the immigration minister special new powers to order non-citizens, who had exhausted all paths to stay in Australia, to take steps to facilitate their own removal.

This would include applying for passports and other steps to attain travel documentation.

Failure to comply, without a reasonable excuse, would constitute a criminal offence with a mandatory minimum of one year in prison, and a maximum sentence of five years.

But the opposition's spokespeople for immigration and home affairs, Dan Tehan and James Paterson, said the Albanese government had failed to explain the urgency of the bill.

"Last night in a specially convened Senate hearing, the government was completely unable to answer basic questions about this legislation and the need for it," Senator Paterson said.

"In fact, Coalition senators basically begged the government to demonstrate why it was so urgent, why it had to be rushed through the Parliament in 36 hours, and they failed to do so."

Officials told senators they had identified a gap in the legislation in February and it needed to be urgently filled.

They said the legislation was not linked to an upcoming High Court case, which concerns non-citizens who remain in detention because they have not cooperated with efforts to remove them from Australia.

"We asked the government last night about what the interaction between this legislation and the High Court was and the upcoming case," Senator Paterson said.

"They were at pains to say there was no connection, that there was no relationship, that it would not influence the case at all.

"We practically begged them to explain there was a link and they failed to provide it."