  Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a
link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in
$M$, in this article, we establish an upper bound of the length of an $n^{th}$
shortest closed geodesic as a logarithmic function of $V$ in $M\setminus L$.
Our works complement the work of Lakeland and Leininger \cite{Christopher} on
the upper bound of systole length.
