State-of-the-art image models predominantly follow a two-stage strategy: pre-training on large datasets and fine-tuning with cross-entropy loss. Many studies have shown that using cross-entropy can result in sub-optimal generalisation and stability. While the supervised contrastive loss addresses some limitations of cross-entropy loss by focusing on intra-class similarities and inter-class differences, it neglects the importance of hard negative mining. We propose that models will benefit from performance improvement by weighting negative samples based on their dissimilarity to positive counterparts. In this paper, we introduce a new supervised contrastive learning objective, SCHaNe, which incorporates hard negative sampling during the fine-tuning phase. Without requiring specialized architectures, additional data, or extra computational resources, experimental results indicate that SCHaNe outperforms the strong baseline BEiT-3 in Top-1 accuracy across various benchmarks, with significant gains of up to $3.32\%$ in few-shot learning settings and $3.41\%$ in full dataset fine-tuning. Importantly, our proposed objective sets a new state-of-the-art for base models on ImageNet-1k, achieving an 86.14\% accuracy. Furthermore, we demonstrate that the proposed objective yields better embeddings and explains the improved effectiveness observed in our experiments.