A partial solution is to remove nimby laws that prevent accessory dwelling units, duplexes, and medium and high density housing. Then you use the tax laws to heavily punish corporate ownership of low and medium density housing, and make it progressively more expensive for mom and pop landlords after a reasonable number of properties.
Hopefully this would lower the cost for single family housing, while increasing supply and variety of rentals. But somehow you’d have to prevent end stage capitalist collusion from fixing prices so competition could actually work to drive down prices.