I'm not an expert on North Carolina research, so I can't help answer a question involving "near Lenoir Co." But I was able to check each city and town in the 1982 book titled, "Adoption Agencies, Orphanages and Maternity Homes--An Historical Directory." Each city and town has its county specified.
Only one entry in this state was in (not near) Lenoir County: Kennedy Home, in Kinston, NC. In 1977 it was called The Kennedy Home of the Baptist Children's Homes of North Carolina, Inc. The book says it opened in 1914, which is after the year mentioned in your query. The Kennedy Home was a branch of the Mills Home in Thomasville.
Your grandfather should be listed in the federal census of 1920, which was indexed by name under the Soundex system. Perhaps it shows the name of the orphanage.