  For an $r$-graph $F$ and integers $n,t$ satisfying $t \le n/v(F)$, let
$\mathrm{ar}(n,tF)$ denote the minimum integer $N$ such that every
edge-coloring of $K_{n}^{r}$ using $N$ colors contains a rainbow copy of $tF$,
where $tF$ is the $r$-graphs consisting of $t$ vertex-disjoint copies of $F$.
The case $t=1$ is the classical anti-Ramsey problem proposed by
Erd\H{o}s--Simonovits--S\'{o}s~\cite{ESS75}. When $F$ is a single edge, this
becomes the rainbow matching problem introduced by Schiermeyer~\cite{Sch04} and
\"{O}zkahya--Young~\cite{OY13}. We conduct a systematic study of
$\mathrm{ar}(n,tF)$ for the case where $t$ is much smaller than
$\mathrm{ex}(n,F)/n^{r-1}$. Our first main result provides a reduction of
$\mathrm{ar}(n,tF)$ to $\mathrm{ar}(n,2F)$ when $F$ is bounded and smooth, two
properties satisfied by most previously studied hypergraphs. Complementing the
first result, the second main result, which utilizes gaps between Tur\'{a}n
numbers, determines $\mathrm{ar}(n,tF)$ for relatively smaller $t$. Together,
these two results determine $\mathrm{ar}(n,tF)$ for a large class of
hypergraphs. Additionally, the latter result has the advantage of being
applicable to hypergraphs with unknown Tur\'{a}n densities, such as the famous
tetrahedron $K_{4}^{3}$.
