Given that Lee's command, the army of Northern Virginia, was only a small portion of the entire Confederate Army despite being the single biggest field army, technically the average soldier would be unlikely to contact Lee in any way.

If you assume the hypothetical soldier was part of the Army of Northern Virginia, then it's most likely that seeing the commander on his horse was the most likely scenario, or perhaps being part of a group of troops that Lee addressed. There are several occasions where he is known to have spoken to brigade and division size formations (1000-4000 men) so perhaps there is a chance for a soldier to have heard him speak. Not very many common soldiers would have had more contact than that, since the Army of Northern Virginia was generally 25,000-60,000 strong, with very frequent "turnover", limiting the proportion of men Lee could have practically had contact with.

We do have many instances in the records of Lee on horseback being recognized sight by bodies of troops and there is even another general officer, William N. Pendleton the chief of artillery, who was often mistaken for Lee by troops. So I think it's safe to say he was commonly observed on horseback by average soldiers, perhaps not even from very far away.