Recent advances in the field of out-of-distribution (OOD) detection have placed great emphasis on learning better representations suited to this task. While there are distance-based approaches, distributional awareness has seldom been exploited for better performance. We present HAC$_k$-OOD, a novel OOD detection method that makes no distributional assumption about the data, but automatically adapts to its distribution. Specifically, HAC$_k$-OOD constructs a set of hypercones by maximizing the angular distance to neighbors in a given data-point's vicinity to approximate the contour within which in-distribution (ID) data-points lie. Experimental results show state-of-the-art FPR@95 and AUROC performance on Near-OOD detection and on Far-OOD detection on the challenging CIFAR-100 benchmark without explicitly training for OOD performance.