Post-selection inference has recently been proposed as a way of quantifying uncertainty about detected changepoints. The idea is to run a changepoint detection algorithm, and then re-use the same data to perform a test for a change near each of the detected changes. By defining the p-value for the test appropriately, so that it is conditional on the information used to choose the test, this approach will produce valid p-values. We show how to improve the power of these procedures by conditioning on less information. This gives rise to an ideal selective p-value that is intractable but can be approximated by Monte Carlo. We show that for any Monte Carlo sample size, this procedure produces valid p-values, and empirically that noticeable increase in power is possible with only very modest Monte Carlo sample sizes. Our procedure is easy to implement given existing post-selection inference methods, as we just need to generate perturbations of the data set and re-apply the post-selection method to each of these. On genomic data consisting of human GC content, our procedure increases the number of significant changepoints that are detected from e.g. 17 to 27, when compared to existing methods.