As a resident of the UK I cannot just wander into the US whenever I fancy. But getting a US visa is not a problem most UK visitors to the US have to worry about because our two fine countries have a ‘visa waiver’ system in place whereby tourists (sans criminal records and not carrying the plague obviously) can fill in a simple form while on board their flight to the US for entry up to a maximum of a 3 month stay. For regular clean-living British folks it’s the only way they ever need to travel.
But hiking the AT obviously takes longer than that to complete. What then? The answer is, you need to get a visa to hike the Appalachian Trail, Pacific Crest Trail or Continental Divide Trail because all three of these National Scenic Trails take more than three months to hike.