The use of approximate solution techniques for the Chemical Master Equation is a common practice for the analysis of stochastic chemical systems. Despite their widespread use, however, many such techniques rely on unverifiable assumptions and only a few provide mechanisms to control the approximation error quantitatively. Addressing this gap, Dowdy and Barton (J Chem Phys 149(7):074103, 2018) proposed an optimization-based technique for the computation of guaranteed bounds on the moment trajectories associated with stochastic chemical systems, thereby providing a general framework for rigorous uncertainty quantification. Here, we present an extension of this method. The key contribution is a new hierarchy of convex necessary moment conditions that build upon partitioning of the time domain. These conditions reflect the temporal causality that is inherent to the moment trajectories associated with stochastic processes described by the Chemical Master Equation and can be strengthened by simple refinement of the time domain partition. Analogous to the original method, these conditions generate a hierarchy of semidefinite programs that furnishes monotonically improving bounds on the trajectories of the moments and related statistics of stochastic chemical systems. Compared to its predecessor, the presented hierarchy produces bounds that are at least as tight and features new bound tightening mechanisms such as refinement of the time domain partition which often enable the computation of dramatically tighter bounds with lower computational cost. We analyze the properties of the presented hierarchy, discuss some aspects of its practical implementation and demonstrate its merits with several examples.