The Maude Swearingen in my family photo, I now believe, is the daughter of Basil and Anna Boyd Swearingen. This makes her the aunt of her namesake, Maude Marie Pugh Falconer (Yula Fay's daughter).
The confusion arose from the notation on the backside of the photo "Aunt Maude, Dad's half sister". The explanation is that Anna Boyd was previously married and had two sons by her first husband. A daughter or son of one of these sons apparently wrote the message. Maude Swearingen died in 1883, not the 1863 in my original query.