Inflationary observables are directly related to the first-passage-time statistics of the stochastic number of e-folds. In some cases, however, these statistics retain an undesirable dependence on the regularising boundary introduced to set the cutoff scale of the effective stochastic description. Although boundary-independent results can be obtained order by order in a small-noise regime, the associated perturbative series are divergent. It is shown that the well-behaved counterparts can nevertheless be extracted from these series through Borel resummation, which also indicates a motivated prescription for introducing the regularising boundary. For a quadratic model, the resummation is directly performed at the level of the full first-passage-time distribution, thereby encompassing all first-passage-time statistics. The asymptotic tail of the distribution is then determined from the analytic structure of its characteristic function.