Considering supercritical Bernoulli percolation on $\mathbb{Z}^d$, Garet and Marchand [GM09] proved a diffusive concentration for the graph distance. In this paper, we sharpen this result by establishing the subdiffusive concentration inequality, which revisits the sublinear bound of the variance proved by Dembin [Dem22] as a consequence. Our approach is inspired by similar work in First-passage percolation [BR08, DHS14], combined with new tools to address the challenge posed by the infinite weight of the model. These tools, including the notion of effective radius and its properties, enable a simple one-step renormalization process as a systematic means of managing the effects of resampling edges.