Part of the procedure for admissions at emergency or otherwise is to ask the question, establish the contact, and have a relationship which pre-exists finding a person. Go back to the ecosystem of folks who provide services to the homeless, including health, and include them in the ecosystem, create the pro forma for the questions you ask. It's the same with law enforcement. If someone is arrested, ask them the question. If someone shows up in court, ask them the question.
The point of contact can be a Veterans Affairs case worker. It could also be part of the ecosystem that is focused on veterans who are in crisis. It could be an SHS. It could be a community service.
The bottom line, though, is that if the community doesn't know who to call, then it's “catch and release”, unfortunately.