  We consider non-monogenic simplest cubic fields $K=\mathbb{Q}(\rho)$ in the
family introduced by Shanks, and among these, we focus in the fields whose
generalized module index $[\mathcal{O}_K:\mathbb{Z}[\rho]]$ is a prime number
$p$. We prove that these fields arise exactly for $p=3$ or
$p\equiv1\,(\mathrm{mod}\,6)$ and we use the method introduced in
arXiv:2005.12312 to find the additive indecomposables of $\mathcal{O}_K$. We
determine the whole structure of indecomposables for the family with $p=3$ and
obtain that the behaviour is not uniform with respect to the indecomposables of
$\mathbb{Z}[\rho]$. From the knowledge of the indecomposables we derive some
arithmetical information on $K$, namely: the smallest and largest norms of
indecomposables, the Pythagoras number of $\mathcal{O}_K$ and bounds for the
minimal rank of universal quadratic forms over $K$.
