BOSTON — About 450 families have left the state’s emergency assistance shelter system since Sept. 1, and the state’s top housing official says there could be momentum in moving even more families out of the strained system.

“We think we can substantially increase that number, which is ultimately the goal,” Housing and Livable Communities Secretary Ed Augustus said on WCVB’s “On the Record” program, which aired Sunday and was filmed Tuesday.

The state’s HomeBASE cash assistance program has helped families grappling with homelessness, he said, but providing resources to new arrivals is “more challenging” since they lack a safety net, such as having friends or family who can assist them with temporary shelter.

About 40 to 50 families are arriving in Massachusetts daily, Augustus said. But Gov. Maura Healey indicated last week the flow of migrants is slowing due to the colder weather.

The secretary said migrants are fleeing “some horrific circumstances,” with the majority of new arrivals coming from Haiti.

“We’re doing the best we can to triage those that arrive,” Augustus said. “Those that are particularly vulnerable, we have kind of capacity for them; others are placed on the waitlist. And as soon as we can exit families from the system, we can bring them in off the waitlist into the shelters.”

Lawmakers failed to strike a deal before Thanksgiving on a bill that’s poised to pump $250 million into the state’s shelter system.

“We desperately need the supplemental budget appropriation that the governor proposed back in September,” Augustus said. “It’s critical to provide the level of service that we’re providing to the 7,500 families that are in the shelter system.”

The system hit Healey’s capacity limit of 7,500 families earlier this month, and 102 families were on the waitlist as of Friday, according to a spokesperson for the Executive Office of Housing and Livable Communities.

Healey has repeatedly called on the federal government to expedite the work authorization process to help migrants who are eager to work, leave shelter and support their families financially.

More than 1,000 migrants received help with their work authorizations during a clinic held earlier this month by the Healey administration with the U.S. Department of Homeland Security. Another clinic is being held this week, and state officials have also connected shelter residents to job training programs.