It was shown that unbounded solutions of the Neumann initial-boundary value problem to the two-dimensional Keller–Segel system can be induced by initial data having large negative energy if the total mass $\mathrm{\Lambda }\in (4\pi ,\mathrm{\infty })\setminus 4\pi \cdot \mathbb{N}$ and an example of such an initial datum was given for some transformed system and its associated energy in Horstmann–Wang (2001). In this work, we provide an alternative construction of nonnegative nonradially symmetric initial data enforcing unbounded solutions to the original Keller–Segel model.