Consider an exact action of a discrete group $G$ on a separable C*-algebra $A$. It is shown that the reduced crossed product $A\rtimes _{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D706}}G$ is strongly purely infinite—provided that the action of $G$ on any quotient $A/I$ by a $G$-invariant closed ideal $I\neq A$ is element-wise properly outer and that the action of $G$ on $A$ is $G$-separating (cf. Definition 5.1). This is the first non-trivial sufficient general criterion for strong pure infiniteness of reduced crossed products of C*-algebras $A$ that are not $G$-simple. In the case $A=\text{C}_{0}(X)$, the notion of a $G$-separating action corresponds to the property that two compact sets $C_{1}$ and $C_{2}$, that are contained in open subsets $C_{j}\subseteq U_{j}\subseteq X$, can be mapped by elements $g_{1},g_{2}$ of $G$ onto disjoint sets $\unicode[STIX]{x1D70E}_{g_{j}}(C_{j})\subseteq U_{j}$, but satisfy not necessarily the contraction property $\unicode[STIX]{x1D70E}_{g_{j}}(U_{j})\subseteq \overline{U_{j}}$. A generalization of strong boundary actions on compact spaces to non-unital and non-commutative C*-algebras $A$ (cf. Definition 7.1) is also introduced. It is stronger than the notion of $G$-separating actions by Proposition 7.6, because $G$-separation does not imply $G$-simplicity and there are examples of $G$-separating actions with reduced crossed products that are stably projection-less and non-simple.