  For $\gamma \geq 7$ and $g \geq 6\gamma + 5$, we construct a family
$\mathcal{F}^{\prime}$ of curves lying on cones in $\mathbb{P}^{g-3\gamma+1}$
over smooth non-degenerate curves of genus $\gamma$ and degree $g-2\gamma$ in
$\mathbb{P}^{g-3\gamma+1}$. We show that $\dim \mathcal{F}^{\prime} =
2g-\gamma-1 + (g-3\gamma+1)^2$. For a general curve $X^{\prime}$ from the
family $\mathcal{F}^{\prime}$, we compute the dimension of the space of its
first-order deformations. We prove that the family $\mathcal{F}^{\prime}$ gives
rise to an irreducible, non-reduced component $\mathcal{D}^{\prime}$ of the
Hilbert scheme $\mathcal{I}_{2g-4\gamma + 1, g, g - 3\gamma + 1}$, which
parametrizes smooth, irreducible, non-degenerate curves of degree $2g-4\gamma +
1$ and genus $g$ in $\mathbb{P}^{g-3\gamma+1}$. We obtain $\dim
T_{[X^{\prime}]} \mathcal{D}^{\prime} = \dim \mathcal{D}^{\prime} + 1 = \dim
\mathcal{F}^{\prime} + 1$.
