Vision Transformers (ViTs) have gained significant attention and widespread adoption due to their impressive performance in various computer vision tasks. However, in practice, their substantial computational overhead often leads to high inference latency and increased overheads when deployed on resource-constrained edge devices like smartphones, autonomous vehicles, and robots. To address these challenges, Early Exit (EE) has emerged as a promising approach for lightweight inference on edge devices. It accelerates inference and reduces computational overhead by adaptively producing predictions through early exits based on sample complexity. Existing EE methods typically suffer from substantial accuracy decreases in late exits while providing only marginal accuracy improvements to early exits. This paper presents EnViT, an exit-aware structured dropout-enabled self-distillation approach that enhances the performance of early exits without compromising late exits. EnViT leverages structured dropout to enable self-distillation, where the full model serves as the teacher and its own virtual sub-models generated by structured dropout as students. This mechanism effectively distills knowledge from the full model to early exits and avoids performance degradation in late exits by mitigating parameter conflicts across exits during training. Evaluation on five datasets shows that our EnViT achieves accuracy improvements ranging from 0.36\% to 7.92\% while maintaining competitive speed-up ratios of 1.72x to 2.23x.