In PA:
The thru hiker bubble is spread across the PA A.T. in June. Blue Rocks campground by Pinnacle/Hawk Mountain is oft cited as a family favorite, though I haven't been there myself, and that area seems relatively crowded for the scenery that doesn't differ much from the rest of PA. Pine Grove Furnace State Park, home to the A.T. museum and near the halfway marker, has some good A.T. hikes and loop back on other trail possibilities. Both this and the nearby Caledonia State Park aren't exactly undiscovered.
Cowans Gap State Park has a nice campground, in an isolated valley without reliable cell service. You'll have to say whether that's good for your family, or not.
It's not extremely far off the PA Turnpike or US 30/other roads. Standing Stone Trail, which offers similar but better scenery to the PA A.T., runs north from here, and Tuscarora Trail north and south, so plenty of Rocksylvania hiking.
In PA, June is a good time for the northcentral "PA Wilds" region. Black flies aren't quite as prevalent as in the far north woods of NY/New England. Worlds End State Park, along Loyalsock Trail and many linked loops, is a highlight not quite as far west as many other possible venues. Little Pine State Park, along Mid State and Tiadaghton Trails, is a less crowded alternative about 1/2 hour drive off I-80. Hills Creek State Park, far enough north along Mid State Trail that it runs out of rocks, has cabins and yurts in addition to campsites. Just north and also along Mid State Trail (but due to road pattern not much different drive time to reach) Ives Run Recreation Area, on US Army Corps of Engineers Hammond Lake, offers a bit more hiking directly from the campsite (but no cabins/yurts).