  Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the
Hilbert scheme whose general points correspond to smooth irreducible
non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. We use
families of curves on cones to show that under certain numerical assumptions
for $d$, $g$ and $r$, the scheme $\mathcal{I}_{d,g,r}$ acquires generically
smooth components whose general points correspond to curves that are double
covers of irrational curves. In particular, in the case $\rho(d,g,r) :=
g-(r+1)(g-d+r) \geq 0$ we construct explicitly a regular component that is
different from the distinguished component of $\mathcal{I}_{d,g,r}$ dominating
the moduli space $\mathcal{M}_g$.
