They already do supplement low wage work via housing support, heating support, food support, health care support. And that still isn't enough!You're describing exactly why I don't support increasing the minimum wage.
I don't think it's time yet for a universal basic income but I would like to see the government supplement low wage work--something along the lines of our earned income tax credit but larger.
How about instead of corporations getting all of this subsidization for their employees from the taxpayer, the corporations pay their workers a wage that no longer requires all these taxpayer supplied subsidies.
I know, I know, costs will go up (and baby Jesus will be bawling). But we are already paying more in taxes because of this, to subsidize corporation pay to their employees!
In other words, off the books welfare. If we are going to provide welfare it should be done openly, not by such subterfuge. Doing it this way just means you pay it through higher prices rather than higher taxes. It's easier to stick your head in the sand about the former but the lack of accounting actually makes it worse.