Demanding tasks often require a series of decisions to reach a goal. Recent progress in perceptual decision-making has served to unite decision accuracy, speed, and confidence in a common framework of bounded evidence accumulation, furnishing a platform for the study of such multi-stage decisions. In many instances, the strategy applied to each decision, such as the speed-accuracy trade-off, ought to depend on the accuracy of the previous decisions. However, as the accuracy of each decision is often unknown to the decision maker, we hypothesized that subjects may carry forward a level of confidence in previous decisions to affect subsequent decisions. Subjects made two perceptual decisions sequentially and were rewarded only if they made both correctly. The speed and accuracy of individual decisions were explained by noisy evidence accumulation to a terminating bound. We found that subjects adjusted their speed-accuracy setting by elevating the termination bound on the second decision in proportion to their confidence in the first. The findings reveal a novel role for confidence and a degree of flexibility, hitherto unknown, in the brain's ability to rapidly and precisely modify the mechanisms that control the termination of a decision.