Well, in a sense, there isn't any land here that isn't owned by anyone. But it gets complicated. First there is the federal government. It owns a lot of land. Some of that is leased but I don't know what rights that gives those who are leasing the land, usually for grazing. Probably all rights, depending on what it was leased for. Some may think it gives them the right to shoot trespassers. Leased land like that is a big issue in some places and it has nothing to do with walking. There are lots of national parks and national forests, which are administered by different departments and have totally different missions and mindsets. The national parts include scenic rivers, national seashores, national monuments and I don't know what all. The Bureau of Land Management has the most land and they are the ones who lease grazing rights.
States have their own parks and forest systems, if any, as well as variations like wildlife management areas, which may have different names in different places. And counties (nothing like an English county) and cities have parks, too, which can sometimes be fairly large and densely wooded.
Everything else is privately owned, which may or may not have easy access or any access at all. For various reasons, I think access to some privately owned land has become more difficult in recent years. By privately owned, I am including vast acreage owned by mining companies and forestry companies, which may seem like unclaimed wilderness but by no means is. Then there is Alaska, which really does have a lot of unclaimed wilderness, though not as much as it used to, before some large parks were created.
As far as I know, there is no privately owned land that is inside a park, although I don't think that is true for national forests. I believe that national forests have as many hiking trails as the larger national parks and the better known long trails, such as the Appalachian Trail in the East, are not entirely within parks, although the actual trail itself might be considered a park. But it does cross private land in a few places as well as going through wildlife management areas. It even goes through at least one town.
