Free housing only solves part of the problem. Homelessness is usually caused by something beyond high rent and addressing that underlying issue(s) is also necessary in the long term. Lack of mental healthcare is a common factor, as are substance abuse (often brought on due to mental health issues) or external debt.

To provide a comprehensive solution, a lot of things are required, such as:

• Free housing for homeless people to provide immediate relief.
• Support for those who are traditionally unemployable. Being homeless and without a "proper" job for a few years makes for a terrible CV; some help is necessary to avoid people falling through the cracks again.
• Better access to mental healthcare, both in terms of cost and availability. Many countries are struggling with one or both of this.
• Destigmatization of mental health issues. People won't seek help if they're too ashamed to admit they have a problem in the first place.
• Likewise, access to and destigmatization of addiction treatments.
• For the United States: Comprehensive healthcare reform to eliminate medical debt as a concept.

The nice thing is that each of those by themselves already help. There's no need to wait for the perfect solution; solving this piecemeal is effective.