The usual confidence interval, based on Student's $t$ distribution, has conditional confidence that is larger than the nominal confidence level. Although this fact is known, along with the fact that increased conditional confidence can be used to improve a confidence assertion, the confidence assertion of Student's $t$ interval has never been critically examined. We do so here, and construct a confidence estimator that allows uniformly higher confidence in the interval and is closer (than $1 - \alpha$) to the indicator of coverage.
"Increasing the Confidence in Student's $t$ Interval." Ann. Statist. 20 (3) 1501 - 1513, September, 1992. https://doi.org/10.1214/aos/1176348781