Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a (random) graph on $n$ vertices obtained by drawing uniformly at random a $d$-regular (simple) graph on $[n]$ and then performing independent $p$-bond percolation on it, i.e. we independently retain each edge with probability $p$ and delete it with probability $1-p$. Let $|\mathcal{C}_{\text{max}}|$ be the size of the largest component in $\mathbb{G}(n, d, p)$. We show that, when $p$ is of the form $p=(d-1)^{-1}(1+λn^{-1/3})$ for $λ\in \mathbb{R}$, and $A$ is large, \begin{align*} \mathbb{P}(|\mathcal{C}_{\text{max}}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3(d-1)(d-2)}{8d^2}+\frac{λA^2(d-2)^2}{2d(d-1)}-\frac{λ^2 A(d-1)}{2(d-2)}}. \end{align*} This improves on a result of Nachmias and Peres. We also give an analogous asymptotic for the probability that a particular vertex is in a component of size larger than $An^{2/3}$.