  We study the finite-time blow-up in two variants of the parabolic-elliptic
Keller-Segel system with nonlinear diffusion and logistic source. In
$n$-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla
\cdot ((u+1)^{m-1}\nabla u - u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 =
\Delta v - \frac1{|\Omega|} \int_\Omega u + u \end{cases} \tag{JL} \end{align*}
and \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -
u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 = \Delta v - v + u,
\end{cases}\tag{PE} \end{align*} where $\lambda$ and $\mu$ are given spatially
radial nonnegative functions and $m, \kappa > 0$ are given parameters subject
to further conditions. In a unified treatment, we establish a bridge between
previously employed methods on blow-up detection and relatively new results on
pointwise upper estimates of solutions in both of the systems above and then,
making use of this newly found connection, provide extended parameter ranges
for $m,\kappa$ leading to the existence of finite-time blow-up solutions in
space dimensions three and above. In particular, for constant $\lambda, \mu >
0$, we find that there are initial data which lead to blow-up in (JL) if
\begin{alignat*}{2} 0 \leq \kappa &< \min\left\{\frac{1}{2}, \frac{n - 2}{n} -
(m-1)_+ \right\}&&\qquad\text{if }
m\in\left[\frac{2}{n},\frac{2n-2}{n}\right)\\ \text{ or }\quad 0 \leq
\kappa&<\min\left\{\frac{1}{2},\frac{n-1}n-\frac{m}2\right\} &&\qquad \text{if
} m\in\left(0,\frac{2}{n}\right), \end{alignat*} and in (PE) if $m \in [1,
\frac{2n-2}{n})$ and \begin{align*} 0 \leq \kappa < \min\left\{\frac{(m-1) n +
1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\}. \end{align*}
