A fresh parliamentary push on same-sex marriage has become more likely after a cross-party Senate committee reached a broad consensus on refining the government's same-sex marriage legislation.

The "unprecedented show of collaboration" paves the way for a marriage equality bill as soon as next month, to be co-sponsored by Labor, the Greens, gay Liberal senator Dean Smith, the NXT and possibly Derryn Hinch.

In the report published Wednesday, senators took aim at aspects of the Marriage Act changes proposed by Attorney-General George Brandis ahead of the failed same-sex marriage plebiscite.

The exposure draft - the first time an Australian government had outlined its vision for marriage equality - proposed sweeping exemptions to discrimination law for ministers of religion, civil celebrants and religious businesses who did not want to participate in same-sex unions.