Dr. Greyhound only handles out of state patients. I think they'll try sending Chris to Dr. Uber first.
And it looks like this only puts Drs. Uber and Greyhound at one step removed. It looks to me like it requires a plan, but not necessarily a good plan, coordinating with the local social services providers (private or public - not specified) to make sure they have a plan to place him somewhere. At least for 30 days. Then the prison can wash their hands of him. So in other words, the prison can't dump him on the streets (for 30 days), but can negotiate a release arrangement that hands him over to another mental health agency that can. Worst case, Chris gets a 30 day reprieve at a temporary mental respite facility before they give him a ticket for Dr. Greyhound (or just open the door and boot him out). They'll probably do their best to place him somewhere, but there's no guarantee they'll succeed.