An $r$-hued coloring of a simple graph $G$ is a proper coloring of its vertices such that every vertex $v$ is adjacent to at least $\min\{r, °(v)\}$ differently colored vertices. The minimum number of colors needed for an $r$-hued coloring of a graph $G$, the $r$-hued chromatic number, is denoted by $χ_{r}(G)$. In this note we show that $$χ_r(G) \leq (r - 1)(Δ(G) + 1) + 2,$$ for every simple graph $G$ and every $r \geq 2$, which in the case when $r < Δ(G)$ improves the presently known $Δ(G)$-based upper bound on $χ_r(G)$, namely $r Δ(G) + 1$. We also discuss the existence of graphs whose $r$-hued chromatic number is close to $(r-1)(Δ+ 1 ) + 2$ and we prove that there is a bipartite graph of maximum degree $Δ$ whose $r$-hued chromatic number is $(r-1)Δ+ 1$ for every $r \in \{2, \dots, 9\}$ and infinitely many values of $Δ\geq r + 2$; we believe that $(r-1)Δ(G) + 1$ is the best upper bound on the $r$-hued chromatic number of any bipartite graph $G$.