There are numerous side hikes all the way along the trail. Which ones you do, if any depends on the goals and objectives of your hike.
In Maine, I managed to do side hikes so that I bagged all but 2 of the 4000 footers in Maine. I would have liked to spend more time in Baxter State Park, but at that point, I was beat. I would have liked to arrange a ride so that I could have gone across the Knife Edge and down into Chimney Pond as one example. My best advice is to look at the AMC Maine Mountain Guide. In fact, I carried the map from this book so I could see what some options for side hikes.
In the White Mountains, there are all kinds of side hikes. When I went through, I tried to bag as many of the 4000 footers as possible. Got 22 out of 46. But, depending on time, there are numberous other places, like waterfalls to side hike to.
The best parts of Shenandoah National Park are off the AT. The maps show all the trails. Maybe brouse some of the books written about hiking in this park for some good hikes. Certainly White Oak Canyon is popular, as is Old Rag, and Camp Hoover, just to name three.
Ditto for Great Smokey Mountains.
You might consider summiting Mt. Rodgers and Whiteface in Virginia.
So many trails (and peaks), so little time.
Peaks