  This article is a continuation of the article with the same title (see
arXiv:1003.2953v1). Let {\rm $\text{HUR}_{d,t}^{G}(\mathbb P^1)$} be the
Hurwitz space of degree $d$ coverings of the projective line $\mathbb P^1$ with
Galois group $\mathcal S_d$ and having fixed monodromy type $t$ consisting of a
collection of local monodromy types (that is, a collection of conjugacy classes
of permutations $\sigma$ of the symmetric group $\mathcal S_d$ acting on the
set $I_d=\{1,...,d\}$). We prove that if the type $t$ contains big enough
number of local monodromies belonging to the conjugacy class $C$ of an odd
permutation $\sigma$ which leaves fixed $f_C\geq 2$ elements of $I_d$, then the
Hurwitz space {\rm $\text{HUR}_{d,t}^{\mathcal S_d}(\mathbb P^1)$} is
irreducible.
