Yes, there are. The only real difference between a mentally ill homeless person and a normal mentally ill person is the home. Why wouldn't a stable home be the answer?
A short-term stay in a mental hospital might be necessary while getting the patient adjusted to their medication. But once someone is stable, there is no need for such things. Even though the institutions have changed, they are still pretty horrible places to stay in, with all sorts of lifestyle restrictions that aren't generally needed.
A small apartment is appropriate for most folks once they are stable, along with access to someone to call. Some communities have social workers that visit once or twice a week and help with grocery shopping, bill paying, and general "how are you" sorts of things. Group homes are another option if someone chooses it: Private space with shared kitchen and whatnot, staffed with someone to help with cooking and cleaning. My ex, diagnosed with schizophrenia, would have been a candidate for such a thing. I was his primary caregiver, fair or not, and his doctor didn't want him living alone for his own protection.
Give support for family members as well - this is something seriously lacking in the US system. He had help, I didn't.
A small amount of folks might need some help for substance abuse, but many short-term mental health facilities are equipped to do this. A fair amount of folks would be helped by police that are capable of helping mentally ill folks with training and a marker on their ID so the cops know what to change in their approach.