There is no reason however, to think that hunting can't supplement a family's food stocks if other sources start to become thin. And there's a lot more to hunt than deer.
There's also more in play than hunting. A family might feed themselves through a combination of any or all of: hunting, trapping, fishing, foraging for wild edibles, home gardening, AND buying from whatever is available on the market.
One interesting thing about this is that those living close to the coasts may be in better shape in one or two regards. Most notably, seafood is one area where "direct to the consumer" sales is a real thing, and there isn't a lot of processing involved. At least where I grew up on the NC coast. It's common for seafood houses to sell oysters, clams, shrimp, etc. directly to consumers.