Multivariate regular variation plays a role assessing tail risk in diverse applications such as finance, telecommunications, insurance and environmental science. The classical theory, being based on an asymptotic model, sometimes leads to inaccurate and useless estimates of probabilities of joint tail regions. This problem can be partly ameliorated by using hidden regular variation [Resnick, 2002, Mitra and Resnick, 2010]. We offer a more flexible definition of hidden regular variation that provides improved risk estimates for a larger class of risk tail regions.