Tangent measure distributions were introduced by \textit{C. Bandt} [Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri (1992)] and \textit{S. Graf} [Monatsh. Math. 120, No. 3-4, 223-246 (1995; Zbl 0841.28011)] as a measure-theoretic tool to describe the local geometry of self-similar sets and measures generated by iteration of contractive similitudes. In this paper, the authors study the tangent measure distributions of a hyperbolic (or cookie-cutter) Cantor set \(C\) generated by certain contractive mappings that are not necessarily similitudes. Let \(\nu\) be the Gibbs measure on \(C\) and let \(\mu\) be the \(s\)-dimensional Hausdorff measure of \(C\), where \(s =\dim_{H} C\). They show that the tangent measure distributions of \(C\), equipped with either \(\nu\) or \(\mu\), are unique almost everywhere and give an explicit formula for these tangent measure distributions in terms of the limit models of \textit{T. Bedford} and \textit{A. M. Fisher} [Proc. Lond. Math. Soc., III. Ser. 64, No. 1, 95-124 (1992; Zbl 0741.28004)].