Let $f$ be a modular form of weight $k$ and Nebentypus $\psi$. By generalizing a construction of Dabrowski and Delbourgo, we construct a $p$-adic $L$-function interpolating the special values of the $L$-function $L(s,{\rm Sym}^2(f)\otimes\xi)$, where $\xi$ is a Dirichlet character. When $s=k-1$ and $\xi=\psi^{-1}$, this $p$-adic $L$-function vanishes due to the presence of a so-called trivial zero. We give a formula for the derivative at $s=k-1$ of this $p$-adic $L$-function when the form $f$ is Steinberg at $p$. If the weight of $f$ is even, the conductor is even and squarefree, and the Nebentypus is trivial this formula implies a conjecture of Benois.