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 Erdős--Simonovits--Sós~\cite{ESS75}. When $F$ is a single edge, this becomes the rainbow matching problem introduced by Schiermeyer~\cite{Sch04} and Özkahya--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 Turán 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 Turán densities, such as the famous tetrahedron $K_{4}^{3}$.