A connected graph $\G$ is called {\em nicely distance--balanced}, whenever there exists a positive integer $γ=γ(\G)$, such that for any two adjacent vertices $u,v$ of $\G$ there are exactly $γ$ vertices of $\G$ which are closer to $u$ than to $v$, and exactly $γ$ vertices of $\G$ which are closer to $v$ than to $u$. Let $d$ denote the diameter of $\G$. It is known that $d \le γ$, and that nicely distance-balanced graphs with $γ= d$ are precisely complete graphs and cycles of length $2d$ or $2d+1$. In this paper we classify regular nicely distance-balanced graphs with $γ=d+1$.