For the Amalfi coast, there are two books that list all of the hikes on the coast and on the islands (one published by Sunflower and the other by Cicerone). I can't wait to go back and park myself somewhere near the tip of the peninsula and just hike. It's a great place to hike because in addition to stunning beauty, there are buses and ferries to get you around to trailheads, and you can pick one base.
I have not done the type of through-hiking you describe, but I have done some hiking on every trip to Italy. When I search locations, the first thing I do in zoom in with Google Maps and look for green areas, then I go to the website of each Italian park I find.
We love the sea, so we especially enjoyed Maremma/Ucellina park in southern Tuscany. South of the Amalfi coast, you might also research Cilento park. We only scratched the surface there, but it's on my return list also. Most recently we did a lot of hiking in le Marche--the options are endless!
Would love to hear more about it if you make it to Umbria, a big gap on my Italy experiences, but I'm sure there are books devoted to hiking there.