  In this paper, we present an almost sure central limit theorem (ASCLT) for
the hyperbolic Anderson model (HAM) with a L\'evy white noise in a
finite-variance setting, complementing a recent work by Balan and Zheng
(\emph{Trans.~Amer.~Math.~Soc.}, 2024) on the (quantitative) central limit
theorems for the solution to the HAM. We provide two different proofs: one uses
the Clark-Ocone formula and takes advantage of the martingale structure of the
white-in-time noise, while the other is obtained by combining the second-order
Gaussian Poincar\'e inequality with Ibragimov and Lifshits' method of
characteristic functions. Both approaches are different from the one developed
in the PhD thesis of C. Zheng (2011), allowing us to establish the ASCLT
without lengthy computations of star contractions. Moreover, the second
approach is expected to be useful for similar studies on SPDEs with
colored-in-time noises, whereas the former, based on It\^o calculus, is not
applicable.
