We have formulated the W3X procedure by incorporating cost-effective post-CCSD(T) components (up to the CCSDT(Q) level) into the W1X-1 protocol, the latter representing a recently reported economical yet accurate approximation to CCSD(T)/CBS. For medium-sized systems, W3X is moderately more computationally demanding than W1X-1, but it is significantly less costly than the W3.2lite and (especially) W3.2 procedures. Because of the use of the cost-effective W1X-1 method as the underlying CCSD(T) component, W3X is also less expensive than the W2.2 protocol, which does not incorporate post-CCSD(T) excitations. We find that, for single-reference systems (the G2/97 set and most of the W4-11 set), W3X is comparable in accuracy to the underlying W1X-1 protocol, as might have been expected. For the more challenging cases of the multireference systems within the W4-11 set, the dissociation of F2 and the automerization of cyclobutadiene, W3X provides improved performance compared with the CCSD(T)-based procedures (W1X-1 and W2.2). Highly multireference chromium oxides CrO, CrO2, and CrO3 are still somewhat challenging for W3X (and even for the higher-level W3.2lite and W3.2 procedures), but the inclusion of the economical post-CCSD(T) terms in W3X already leads to a significant improvement over W1X-1. Thus, W3X provides a cost-effective means for treating systems with significant (but perhaps not excessive) multireference character that are otherwise not well-described by CCSD(T)-based methods.