There are a lot of variable's
- Season: Summer has more sun then other times of the year
- Direction of travel: The charger needs to face direct sun to be most effective, You can't just put it on you back and leave it there.
- Cover: More tree cover = Less sun
- Budget: More efficient solar chargers tend to be more expensive
- Resupply: How often will you be able to get new batteries and how many will you need to carry.
Luckily for your question about the Appalachian Trail, the answer is simple. A solar charger never becomes more weight-efficient, a couple of references from experienced hikers.
Throughout her adventures, she has attempted to use a solar charger a number of times. On the AT, Kelley sent her charging system home after the first 30 miles when she realized the tree cover wouldn’t allow for enough direct sunlight. Source
The nickname of the AT is the green tunnel. Why? Because you are in the forest. Not direct sunshine. Get a power pack not a solar charger. Source
As an alternative google 'shoes battery charger' there are a number of hits for 'piezoelectric transducer' they convert physical energy into AC electricity, you can charge your devices while hiking sun or no sun.