  We consider quadratic Weyl sums $S_N(x;c,\alpha)=\sum_{n=1}^N\exp\{2\pi
i((\frac{1}{2}n^2+cn)x+\alpha n)\}$ for $c=\alpha=0$ (the rational case) or
$(c,\alpha)\notin\mathbb{Q}^2$ (the irrational case), where $x$ is randomly
distributed according to a probability measure absolutely continuous with
respect to the Lebesgue measure. The limiting distribution in the complex plane
of $\frac{1}{\sqrt{N}}S_N(x;c,\alpha)$ as $N\to\infty$ was described by Marklof
[13] (respectively Cellarosi and Marklof [5]) in the rational (resp.
irrational) case. According to the limiting distribution, the probability of
landing outside a ball of radius $R$ is known to be asymptotic to $\frac{4\log
2}{\pi^2}R^{-4}(1+o(1))$ in the rational case and to
$\frac{6}{\pi^2}R^{-6}(1+O(R^{-12/31}))$ in the irrational case, as
$R\to\infty$. In this work we refine the technique of Cellarosi and Marklof [5]
to improve the known tail estimates to $\frac{4\log
2}{\pi^2}R^{-4}(1+O_\varepsilon(R^{-2+\varepsilon}))$ and
$\frac{6}{\pi^2}R^{-6}(1+O_\varepsilon(R^{-2+\varepsilon}))$ for every
$\varepsilon>0$. In the rational case, we rely on the equidistribution of a
rational horocycle lift to a torus bundle over the unit tangent bundle to the
classical modular surface. All the constants implied by the
$O_\varepsilon$-notations are made explicit
