  We study the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ parametrizing
smooth, irreducible, non-degenerate and linearly normal curves of degree $d$
and genus $g$ in $\mathbb{P}^r$ whose complete and very ample hyperplane linear
series $\mathcal{D}$ have relatively small index of speciality
$i(\mathcal{D})=g-d+r$. In particular we show the existence (and non-existence
as well in some sporadic cases) of every Hilbert scheme of linearly normal
curves with $i(\mathcal{D})=4$. We also determine the irreducibility of
$\mathcal{H}^\mathcal{L}_{2r+4,r+8,r}$ for $3\le r\le 8$, which are rather
peculiar families in a certain sense.
