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

Addendum to "Contact stationary Legendrian surfaces in $\mathbb{S}^5$"[Pacific Math. J. 293(2018), no.1, 101-120]