Improved point and interval estimation of the smallest scale parameter of n independent populations following two-parameter exponential distributions are studied. The model is formulated in such a way that allows for treating the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The classes of improved point and interval estimators are enriched with Stein-type, Brewster and Zidek-type, Maruyama-type and Strawderman-type improved estimators under both quadratic and entropy losses, whereas using as a criterion the coverage probability, with Stein-type, Brewster and Zidek-type, and Maruyama-type improved intervals. The sampling framework considered incorporates important life-testing schemes such as i.i.d. sampling, type-II censoring, progressive type-II censoring, adaptive progressive type-II censoring, and record values.