Amazingly, this USA story had it right. Long distance hikers on trails do love the Post Office. We come to view it as standard procedure for shipping transitional supplies to ourselves. No one is needed as a support person back home if we do this via Bounce Boxes.
Read this article for a very realistic picture of how it goes, ideally.
There are places on the trail that have no readily accessible post offices. At those times, a person knows they'll be resupplying at the nearest gas station for enough to make it to a decent road where they can hitchhike to a nearby town. Some hitch hikes involve 20 miles in one direction. You want to choose a town on a busy road or you could be standing there a very long time.
There are trail towns with hiker hostels nearby. Sometimes the owners will let you mail yourself a package with their address on it. Vermillion Valley, on John Muir Trail/Pacific Crest Trail, and the Sauffley's in Aqua Dulce are just two examples. Every year these places can change their policies so its always wise to check first as you plan your hike.
There are motels along the highways which will allow you to mail a package to yourself if you have a reservation with them. I saw several hikers making up resupply boxes to ship to themselves at these places instead of post offices mainly because the motels are open 24-7 whereas tiny post offices had unusual hours.
No doubt long distance hikers and section hikers will determine these key spots and everything will go on just fine. The Pacific Crest Trail has a dedicated website and there are many forums for Appalachian Trail questions.
If you chose to mail yourself extra clothing to get through the Whites on the AT, or through the John Muir higher elevations, if you can ship something that you are willing to donate to the hiker boxes at the end of that section, you can eliminate the need to ship it back home. North and South bounding hikers on the AT can take advantage of these hiker boxes and generosity much easier than those on the PCT because the traffic and common towns are much fewer on the PCT.