I just realized I never added this information that I got from a retired professor at NC State’s forestry school. He wrote the following to me about the tree I asked about in this thread:
You have the terminology related to bud scales reversed. Bud scales that overlap (like on pignut or shagbark) are called imbricate (overlapping like shingles on a roof). Bud scales that do not overlap (as on pecan) are valvate (like the halves of a [bivalve] clam shell).
From the pictures you provide, I favor Carya ovalis, false shagbark hickory, for the identity of your mystery tree, based upon the bark, bud and fruit. I concur with you; the fruit husk is too thin and the buds too small for shagbark hickory.
In my opinion, pignut and false shagbark hickory are very closely related, and possibly not worthy of recognition as separate species, an opinion I share with others. I do not agree with your reference. Pignut typically has 5 leaflets, and false shagbark has both 5 and 7 leaflets with the majority of leaves having 5 leaflets. Major clues for me were the nearly spherical shape of the fruit, the thin husk and the narrow and short shags of the bark, not the long (2-3 ft) and wide (4-6 ins) shags of mature shagbark. Also recall that shagbark leaves are very large, 16-24 inches long.
…Hickories (like many trees) hybridize, and the wild trees in your woods may help. But a second shagbark would provide better results. Shagbark is known to hybridize with bitternut, pecan and shellbark, but the other hickory species are a possibility.