  Let $C$ be a smooth curve which is complete intersection of a quadric and a
degree $k>2$ surface in $\mathbb{P}^3$ and let $C^{(2)}$ be its second
symmetric power. In this paper we study the finite generation of the extended
canonical ring $R(\Delta,K) :=
\bigoplus_{(a,b)\in\mathbb{Z}^2}H^0(C^{(2)},a\Delta+bK)$, where $\Delta$ is the
image of the diagonal and $K$ is the canonical divisor. We first show that
$R(\Delta,K)$ is finitely generated if and only if the difference of the two
$g_k^1$ on $C$ is torsion non-trivial and then show that this holds on an
analytically dense locus of the moduli space of such curves.
