  In this paper, we give a new construction of $u_0\in B^\sigma_{p,\infty}$
such that the corresponding solution to the hyperbolic Keller-Segel model
starting from $u_0$ is discontinuous at $t = 0$ in the metric of
$B^\sigma_{p,\infty}(\R^d)$ with $d\geq1$ and $1\leq p\leq\infty$, which
implies the ill-posedness for this equation in $B^\sigma_{p,\infty}$. Our
result generalizes the recent work in \cite{Zhang01} (J. Differ. Equ. 334
(2022)) where the case $d=1$ and $p=2$ was considered.
