Productivity issues such as lengthy compilation and limited code reuse have restricted usage of field-programmable gate arrays (FPGAs), despite significant technical advantages. Recent work into overlays-virtual coarse-grained architectures implemented atop FPGAs-has aimed to address these concerns through abstraction, but have mostly focused on pipelined applications with minimal control requirements. Although research has introduced overlays for finite-state machines, those architectures suffer from limited scalability and flexibility, which we address with a new overlay architecture using memory decomposition on transitional logic. Although our overlay provides modest average improvements of 15% to 29% fewer lookup tables for individual finite-state machines, for the more common usage of an overlay supporting different finite-state machines, our overlay achieves a 77% to 99% reduction in lookup tables. In addition, our overlay reduces compilation time to tenths of a second to enable rapid iterative-development methodologies.