  In this paper, we present a new record for the densest geodesic congruent
ball packing configurations in $\mathbf{H}^2\!\times\!\mathbf{R}$ geometry,
generated by screw motion groups. These groups are derived from the direct
product of rotational groups on $\mathbf{H}^2$ and some translation components
on the real fibre direction $\mathbf{R}$ that can be determined by the
corresponding Frobenius congruences. Moreover, we developed a procedure to
determine the optimal radius for the densest geodesic ball packing
configurations related to the considered screw motion groups. The highest
packing density, $\approx0.80529$, is achieved by a multi-transitive case given
by rotational parameters $(2,20,4)$. E. Moln\'{a}r demonstrated that
homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere
$\mathcal{PS}^3(\mathbf{V}^4, \boldsymbol{V}_4, \mathbf{R})$. We use this
projective model of $\mathbf{H}^2\!\times\!\mathbf{R}$ to compute and visualize
the locally optimal geodesic ball arrangements.
