Models for extreme values are generally derived from limit results, which are meant to be good enough approximations when applied to finite samples. Depending on the speed of convergence of the process underlying the data, these approximations may fail to represent subasymptotic features present in the data, and thus may introduce bias. The case of univariate maxima has been widely explored in the literature, a prominent example being the slow convergence to their Gumbel limit of Gaussian maxima, which are better approximated by a negative Weibull distribution at finite levels. In the context of subasymptotic multivariate extremes, research has only dealt with specific cases related to componentwise maxima and multivariate regular variation. This paper explores the conditional extremes model (Heffernan and Tawn, 2004) in order to shed light on its finite-sample behaviour and to reduce the bias of extrapolations beyond the range of the available data. We identify second-order features for different types of conditional copulas, and obtain results that echo those from the univariate context. These results suggest possible extensions of the conditional tail model, which will enable it to be fitted at less extreme thresholds.