We show that, when the double bootstrap is used to improve performance of bootstrap methods for bias correction or constructing confidence intervals, techniques based on using a single double-bootstrap sample for each single-bootstrap sample can be particularly effective. In particular, they produce third-order accuracy for much less computational expense than is required by conventional doublebootstrap methods. However, the results are a little different in the cases of bias correction and confidence interval construction or distribution estimation. For example, in the setting of bias correction there are still theoretical advantages to using relatively large numbers of simulations in the second stage, even though the order of magnitude of error is not affected. This is not true when constructing a confidence interval or estimating a distribution function.