Nyssa sylvatica? If so, the fruit will turn almost black.

Here is a link that might be useful: black gum

Nyssa sylvatica has alternate leaves.

This is Chinese privet.

Forestiera ligustrina or similar sp.?

Here is a link that might be useful: Forestiera ligustrina

Well, I suppose that could be. A noticeable difference from privet is having the flowers/fruit in leaf axils along the branch whereas privet has them at the ends (terminal).

USDA does show it as being found in SC.