For San Francisco in particular, if you work in San Francisco, you pay a local income tax of ~0.4%. Make having paid either that (say, $50 paid by either you or the person you're married to) or having attended an SFUSD school as qualifiers for assistance.

That covers the large majority of homeless that SF should target for help and are all easily accessible to the city government. You can imagine corner cases (attended a private school, lived in SF but worked outside it) but it'd be an improvement over the status quo. Even many of the corner cases could be captured with additional documentation that would require a bit more effort on the applicant's side.