Speaking to some of the points made in general, one of the hurdles here even in the rural South with access is a result of the general trend of people moving to urban and suburban areas and losing that "connection with the land," while rural land becomes more consolidated to agriculture, larger private properties owned by fewer owners, etc.
The displacement of a more generalist type of hunter also came with the massive growth in interest in trophy whitetail hunting. (In a lot of ways, what's happening to western hunting now happened to hunting east of the Mississippi 20 years ago.)
Lots of people lost access to land they used for running raccoons, hunting rabbits, etc. around that time. Before then, even a more urban population could find casual access to private land. When hunting got "serious" so to speak, whitetail hunting sort of took the throne and people got a lot more covetous of their land.
So 20 years later and the problem of getting more of the public onto the private down here is a difficult one. Here in Georgia, to really open up opportunities, especially in the southern half of the state I think would take some sort of block management style incentive.
I have no idea what that system would cost or entail or how feasible it is. A grassroots organization would potentially be a better option.