Statistical models for extreme values are generally derived from non-degenerate probabilistic limits that can be used to approximate the distribution of events that exceed a selected high threshold. If convergence to the limit distribution is slow, then the approximation may describe observed extremes poorly, and bias can only be reduced by choosing a very high threshold at the cost of unacceptably large variance in any subsequent tail inference. An alternative is to use sub-asymptotic extremal models, which introduce more parameters but can provide better fits for lower thresholds. We consider this problem in the context of the Heffernan–Tawn conditional tail model for multivariate extremes, which has found wide use due to its flexible handling of dependence in high-dimensional applications. Recent extensions of this model appear to improve joint tail inference. We seek a sub-asymptotic justification for why these extensions work and show that they can improve convergence rates by an order of magnitude for certain copulas. We also propose a class of extensions of them that may have wider value for statistical inference in multivariate extremes.