A general procedure for multistage modification of pivotal statistics is developed to improve the normal approximation. Bootstrapping a first stage modified statistic is shown to be equivalent, in terms of asymptotic order, to the normal approximation of a second stage modification. Explicit formulae are given for some basic cases involving independent random samples and samples drawn without replacement. The Hodges-Lehmann deficiency is calculated to compare the regular $t$-statistic with its one-step correction.
"Edgeworth Corrected Pivotal Statistics and the Bootstrap." Ann. Statist. 13 (1) 116 - 132, March, 1985. https://doi.org/10.1214/aos/1176346580