  This paper deals with a hyperbolic Keller-Segel system of consumption type
with the logarithmic sensitivity \begin{equation*}
  \partial_{t} \rho = - \chi\nabla \cdot \left (\rho \nabla \log c\right),\quad
\partial_{t} c = - \mu c\rho\quad (\chi,\,\mu>0) \end{equation*} in
$\mathbb{R}^d\; (d \ge1)$ for nonvanishing initial data. This system is closely
related to tumor angiogenesis, an important example of chemotaxis. We firstly
show the local existence of smooth solutions corresponding to nonvanishing
smooth initial data. Next, through Riemann invariants, we present some
sufficient conditions of this initial data for finite-time singularity
formation when $d=1$. We then prove that for any $d\ge1$, some nonvanishing
$C^\infty$-data can become singular in finite time. Moreover, we derive
detailed information about the behaviors of solutions when the singularity
occurs. In particular, this information tells that singularity formation from
some initial data is not because $c$ touches zero (which makes $\log c$
diverge) but due to the blowup of $C^1\times C^2$-norm of $(\rho,c)$. As a
corollary, we also construct initial data near any constant equilibrium state
which blows up in finite time for any $d\ge1$. Our results are the extension of
finite-time blow-up results in \cite{IJ21}, where initial data is required to
satisfy some vanishing conditions. Furthermore, we interpret our results in a
way that some kinds of damping or dissipation of $\rho$ are necessarily
required to ensure the global existence of smooth solutions even though initial
data are small perturbations around constant equilibrium states.
