As a rule, the only single statistic that is clearly correlated with homelessness is the delta between the prevailing wage and the prevailing rent in a geographic area. You can basically predict a city's homelessness rate by the affordability of housing. Sure, people try and cite all kinds of other things. They'll say it has to do with weather, with mental health, with drug use, with available social services. But there's also statistics that basically confound these kinds of measures so badly you can reasonably claim them to be falsified. Homelessness is, above all else, an effect of housing unaffordability. Anyone who talks about the issue who tries to pretend ANY other issue matters more than lack of affordable housing is suspect. You may or may not think this sounds insightful, but the truth is that the gulf between prevailing wages and prevailing rent is really just a measure of how painful it is to get a residence in an area -- so of COURSE homelessness increases with it.
Meanwhile, MOST US cities have laws, rules, and processes set up that reduce housing supply, either intentionally or as an obvious consequence. Policies that objectively make no sense.
Labyrinthine and unreasonable permitting processes. The slow death squeeze of the local builder by national firms that churn out giant, expensive, ticky-tacky homes. Redlining-era restrictive zoning rules that prevent sensible infill. Mandatory parking minimums based on voodoo 'science' that can make it flatly impossible to build dense housing (even in bikeped or transit-connected neighborhoods). The gradual death of the local bank, forcing ALL projects to fit standard national finance paper products that may not work or make sense in your city. Lack of regulation on the rental market or incentivization of owner-occupancy. Lack of said bikeped and transit connections, which de facto increases costs by forcing residents to own, maintain, and use cars for all their trips.
I absolutely think labor organization is a big way to fight back against homelessness because improving bottom-line prevailing wages is going to be a way to fight back against homelessness -- but no single solution will do the job on its own. And most of the best solutions for this can actually start super local -- they don't need national or even state elections to go a certain way, they just need local politicians to listen. You, as an individual, can email, call, or show up to MPO/council meetings and move the needle on a lot of these.
If you want to, you can start with something like the Strong Towns action lab to view guides, advice, and script-like documents for what to actually do and say at these meetings.