I see that Sol May has a military service record. Have you already seen that? He served in Company B of the 39th Kentucky Infantry (mounted). I also see him in the Descriptive list of Kentucky enlistments, which although its title contains the word "colored," contains thousands of non-Black enlistments too, including many White (non-Black, non-slave) union enlistments. Some of the Black people on the list were free too, not all slaves-- in those cases it is indicated on the list.
From those two sources it is evident that Sol lived as a White man. This is not to say that he could not have been "passing", meaning he might have had an African American ancestor, but he/she was far enough removed that he was able to pass for a White man if so. If he had not been considered White, he would have been enlisted into one of the U.S. regiments, as opposed to the state regiments. Further, he enlisted inside of Kentucky, within the county of his birth, so he was also not a runaway when he presented himself as "White" at enlistment. This is just more confirmation that even if Sol was partially Black, he was not enslaved at that time. Now, just in general, as for the possibility of him potentially being both a slave and 100% white-ancestry (i.e. born of a white woman), we can eliminate that one because that was not legal in Kentucky in his lifetime. However, one of his parents could always have been a mulatto (part-Black) slave, and maybe his parentage was kept secret. But you would only be able to find that out by identifying his parents, which I see you are trying hard to do! My advice as a starting point, would be for you to try to get a copy of his full pension record from the National Archives. I see that an Invalid pension (1895) application and Widow's pension application (1923) are both on file for him. (looks like he died Feb 10, 1923 in Norton, VA). It is expensive, but if he has a pension application on file, it can sometimes contain hundreds of pages of family information. I wish I lived close enough to go check for you. But meanwhile best of luck in your search!