We consider, for $h,E>0$, the semiclassical Schrödinger operator $-h^2\Delta + V - E$ in dimension two and higher. The potential $V$, and its radial derivative $\partial_{r}V$ are bounded away from the origin, have long-range decay and $V$ is bounded by $r^{-\delta}$ near the origin while $\partial_{r}V$ is bounded by $r^{-1-\delta}$, where $0\leq\delta < 4(\sqrt{2}-1)$. In this setting, we show that the resolvent bound is exponential in $h^{-1}$, while the exterior resolvent bound is linear in $h^{-1}$.