  In \cite{Luo}, the present author proved that if $L$ is a contact stationary
Legendrian surface in $\mathbb{S}^5$ with the canonical Sasakian structure and
the square length of its second fundamental form belongs to $[0,2]$. Then we
have that $L$ is either totally umbilical or is a flat minimal Legendrian
torus. In this addendum we further prove that if $L$ is a totally umbilical
contact stationary Legendrian surface in $\mathbb{S}^5$ , then $L$ is totally
geodesic.
