  We study the Hilbert scheme of smooth, irreducible, non-degenerate and
linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ ($r\ge 3$)
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
determine the existence as well as the non-existence of Hilbert schemes of
linearly normal curves $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ for every possible
triples $(d,g,r)$ with $i(\mathcal{D})=5$ and $r\ge 3$. We also determine the
irreducibility of the Hilbert scheme $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$
when the genus $g$ is near to the minimal possible value with respect to the
dimension of the projective space $\mathbb{P}^r$ for which
$\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, say $r+9\le g\le r+11$.
In the course of proofs of key results, we show the existence of linearly
normal curves of degree $d\ge g+1$ with arbitrarily given index of speciality
with some mild restriction on the genus $g$.
