An Anker solar panel and the smallest Anker battery worked great for me on the JMT. But out in the Sierra mountains, I often had full sun above tree line, or a thin tree cover compared to the 'Green tunnel' of the AT.
But based on some testing I did on my trip, I would think that it would be possible to use a solar panel, even in the 'Green Tunnel'. But you would have to use one of the bigger panels capable of putting out over 10 Watts. In that size category, the best panel I can think of from a combination of cost, durability/quality, and weight will be the
Anker Solar Lite. The combination of this panel and an
Anker 'lipstick' battery will weigh about 1 pound.
Depending upon the density of the trees, a solar panel of this size will, over time, manage to get some charge to the battery. You would just need to augment having the panel on the back of your pack with taking any opportunity you stop for a rest to put the unit in full sun. Don't expect to be able to use your phone full-time (such as a GPS unit) with this sort of setup. But compared to relying only on a spare battery, with a larger solar panel, you would at least be able to use something like your phone at least when ever you have a clear view of the sky.