If we magic in twice as many housing units there would be calamity and chaos, because a bunch of building magically appeared, in violation of the understood laws of physics. I doubt money would be as much of an important concern in the ensuing chaos. So sure, if we look at a supply and demand graph, and double the supply, the intersection of the two moves, meaning the price goes down.
However, we don't live in a world of perfectly spherical cows and because we don't live in a world where we can double housing units overnight, realistically the reasons we're in this problem is because we can't build enough housing units. Given than, we're not going to be able to double the number of housing units, so realistically we can't build enough to lower rents.
The reasons we can't build enough housing aren't going away overnight, so even pretending they did, it would still take many months, if not years, before new housing units were being sold. Given the time it would take to double housing supply, the natural increase in demand over time because the population is increasing means it would rise to meet supply to keep rents at the current level. Lets say demand increases by 9% each year and it takes 10 years for double the housing stock to come online. Demand is basically doubled by the end of 10 years if that 9% remains constant.
With those problems not disappearing, new housing units are going to be slow to come online, so realistically, again, realistically, because we're not going to double the housing supply, so the best (and my bias is that of being in the SF Bay Area) we can hope for is for rents to increase at a slower rate. (Aka the derivative of rents goes down.)
I just don't seeing the rents themselves decreasing, relative to inflation of money, as well as inflation of the population.