Factorization semigroups and irreducible components of the Hurwitz space. II

We continue the investigation started in [1]. Let $\mathrm{HUR}_{d,t}^{\mathcal S_d}(\mathbb P^1)$ be the Hurwitz space of coverings of degree $d$ of the projective line $\mathbb P^1$ with Galois group $\mathcal S_d$ and monodromy type $t$. The monodromy type is a set of local monodromy types, which are defined as conjugacy classes of permutations $\sigma$ in the symmetric group $\mathcal S_d$ acting on the set $I_d=\{1,\dots,d\}$. We prove that if the type $t$ contains sufficiently many local monodromies belonging to the conjugacy class $C$ of an odd permutation $\sigma$ which leaves $f_C\geqslant 2$ elements of $I_d$ fixed, then the Hurwitz space $\mathrm{HUR}_{d,t}^{\mathcal S_d}(\mathbb P^1)$ is irreducible.