Conformal prediction has attracted significant attention as a distribution-free method for uncertainty quantification in black-box models, providing prediction sets with guaranteed coverage. However, its practical utility is often limited when these prediction sets become excessively large, reducing its overall effectiveness. In this paper, we introduce a novel approach to conformal prediction for classification problems, which leverages a multi-dimensional nonconformity score. By extending standard conformal prediction to higher dimensions, we achieve better separation between correct and incorrect labels. Utilizing this we can focus on regions with low concentrations of incorrect labels, leading to smaller, more informative prediction sets. To efficiently generate the multi-dimensional score, we employ a self-ensembling technique that trains multiple diverse classification heads on top of a backbone model. We demonstrate the advantage of our approach compared to baselines across different benchmarks.