Follow up with more pictures. I think persimmon. Not hairy under hand lens. Upper leaf darker than under. No berries I could see, but seem to recall seeing little white flowers in spring. Is in shaded area near to a dogwood, hickory and sourwood tree. Maybe twenty feet.
thank you for the follow-up. Based on your leaf pictures this does look like a persimmon, Diospyros virginiana. I am still not comfortable with the bark picture you sent, because persimmon bark is usually more deeply furrowed – you may want to check that on the internet. That, perhaps, could vary with age. But the venation and lack of hairs, along with the alternate leaf arrangement and associates do seem to agree with a persimmon. The flowers are very distinctive if you see it in flower next spring.
Dr. Steven Hill, Botanist, SCNPS