  The anti-Ramsey number $\mathrm{ar}(n,F)$ of an $r$-graph $F$ is the minimum
number of colors needed to color the complete $n$-vertex $r$-graph to ensure
the existence of a rainbow copy of $F$. We prove a general upper bound for
$\mathrm{ar}(n,F)$ when $F$ is the expansion of a hypergraph with smaller
uniformality, which refines the general bound $\mathrm{ar}(n,F) =
\mathrm{ex}(n,F_{-}) + o(n^r)$ by Erd{\H o}s--Simonovits--S{\' o}s. Here
$F_{-}$ is the family of $r$-graphs obtained from $F$ by removing one edge. We
also determine the exact value of $\mathrm{ar}(n,F)$ for large $n$ when $F$ is
the expansion of a complete graph, extending a result of Erd{\H
o}s--Simonovits--S{\' o}s from graphs to hypergraphs.
