  In this paper we consider curves on a cone that pass through the vertex and
are also triple covers of the base of the cone, which is a general smooth curve
of genus $\gamma$ and degree $e$ in $\mathbb{P}^{e-\gamma}$. Using the free
resolution of the ideal of such a curve found by Catalisano and Gimigliano, and
a technique concerning deformations of curves introduced by Ciliberto, we show
that the deformations of such curves remain on cones over a deformation of the
base curve. This allows us to prove that for $\gamma \geq 3$ and $e \geq
4\gamma + 5$ there exists a non-reduced component $\mathcal{H}$ of the Hilbert
scheme of smooth curves of genus $3e + 3\gamma$ and degree $3e+1$ in
$\mathbb{P}^{e-\gamma+1}$. We show that $\dim T_{[X]} \mathcal{H} = \dim
\mathcal{H} + 1 = (e - \gamma + 1)^2 + 7e + 5$ for a general point $[X] \in
\mathcal{H}$.
