  We consider Walsh's conformal map from the exterior of a set
$E=\bigcup_{j=1}^\ell E_j$ consisting of $\ell$ compact disjoint components
onto a lemniscatic domain. In particular, we are interested in the case when
$E$ is a polynomial preimage of $[-1,1]$, i.e., when $E=P^{-1}([-1,1])$, where
$P$ is an algebraic polynomial of degree $n$. Of special interest are the
exponents and the centers of the lemniscatic domain. In the first part of this
series of papers, a very simple formula for the exponents has been derived. In
this paper, based on general results of the first part, we give an iterative
method for computing the centers when $E$ is the union of $\ell$ intervals.
Once the centers are known, the corresponding Walsh map can be computed
numerically. In addition, if $E$ consists of $\ell=2$ or $\ell=3$ components
satisfying certain symmetry relations then the centers and the corresponding
Walsh map are given by explicit formulas. All our theorems are illustrated with
analytical or numerical examples.
