In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call $\mathrm{\Sigma }$-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are $\mathrm{\Sigma }$-Prikry. We showed that given a $\mathrm{\Sigma }$-Prikry poset $\mathbb{P}$ and a $\mathbb{P}$-name for a non-reflecting stationary set $T$, there exists a corresponding $\mathrm{\Sigma }$-Prikry poset that projects to $\mathbb{P}$ and kills the stationarity of $T$. In this paper, we develop a general scheme for iterating $\mathrm{\Sigma }$-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.