The government’s controversial changes to higher education fees now appear set to pass the Senate, with Centre Alliance giving its support.

The minor party, whose two federal parliamentarians come from South Australia, has won modest concessions, including 12,000 extra places for students in SA, in return for agreeing to back the bill.

Centre Alliance now has only one Senate crossbencher, Stirling Griff, whose vote will be crucial to get the legislation across the line.

The revamp of fees will mean a major rise in what students have to pay for some courses, including the humanities and law, but reduce the student cost of courses such as nursing and teaching.

The government says the new structure will provide incentives for students to choose courses which are “more job-relevant”.

Pauline Hanson’s One Nation will vote for the changes, but crossbenchers Jacqui Lambie and Rex Patrick are opposed.

Patrick, an independent who is formerly from Centre Alliance, attacked that party’s education spokeswoman and member for Mayo, Rebekha Sharkie, who negotiated with the government.

After Sharkie said on Twitter she would be “forever grateful” for her arts degree, Patrick tweeted: “So, whilst you are forever grateful for the opportunity afforded you, you don’t care for future students in your electorate or state that might want the same opportunity.”

The Senate debates the bill on Tuesday, but it is not clear when the vote will take place. If it is not this week, the next opportunity would be in November. The new fees regime is due to start next year.

Sharkie said the reforms would “encourage universities to strengthen industry relationships and produce job-ready graduates”.

The changes have won support in principle from most universities, with calls for specific alterations. But critics attack the bias against the humanities and dispute the government’s claims about the number of new places that will be created.