Let $\mathcal{B}$ be a $\sigma$-unital $C^*$-algebra. We show that every strongly continuous $E_0$-semigroup on the algebra of adjointable operators on a full Hilbert $\mathcal{B}$-module $E$ gives rise to a full continuous product system of correspondences over $\mathcal{B}$. We show that every full continuous product system of correspondences over $\mathcal{B}$ arises in that way. If the product system is countably generated, then $E$ can be chosen countably generated, and if $E$ is countably generated, then so is the product system. We show that under these countability hypotheses there is a one-to-one correspondence between $E_0$-semigroups up to stable cocycle conjugacy and continuous product systems up to isomorphism. This generalizes the results for unital $\mathcal{B}$ to the $\sigma$-unital case.