  This letter studies the Euler-alignment system with weakly singular influence
functions by introducing a novel technique to bound the density. Instead of
resorting to a nonlinear maximum principle used in [C. Tan, Nonlinearity, 33:
1907--1924, 2020] to bound the interaction term $\psi\ast\rho$ by $\rho^s$ with
$s\in (0, 1)$ for $\psi$ with an algebraic singularity at origin, we bound
$\psi*\rho$ by a relaxed constant for any $\psi \in L^1$. We thus establish the
global-in-time existence results with weaker assumptions on $\psi$ and refined
solution bounds, characterized by the structure of $\psi$.
