We consider optimal control of a multi-class queue in the Halfin--Whitt regime, and revisit the notion of asymptotic optimality and the associated optimality gaps. The existing results in the literature for such systems provide asymptotically optimal controls with optimality gaps of $o(\sqrt{n})$ where $n$ is the system size, for example, the number of servers. We construct a sequence of asymptotically optimal controls where the optimality gap grows logarithmically with the system size. Our analysis relies on a sequence of Brownian control problems, whose refined structure helps us achieve the improved optimality gaps.