We consider the problem of configuring general-purpose solvers to run efficiently on problem instances drawn from an unknown distribution, a problem of major interest in solver autoconfiguration. Following previous work, we focus on designing algorithms that find a configuration with near-optimal expected capped runtime while doing the least amount of work, with the cap chosen in a configuration-specific way so that most instances are solved. In this paper we present a new algorithm, CapsAndRuns, which finds a near-optimal configuration while using time that scales (in a problem dependent way) with the optimal expected capped runtime, significantly strengthening previous results which could only guarantee a bound that scaled with the potentially much larger optimal expected uncapped runtime. The new algorithm is simpler and more intuitive than the previous methods: first it estimates the optimal runtime cap for each configuration, then it uses a Bernstein race to find a near optimal configuration given the caps. Experiments verify that our method can significantly outperform its competitors.