  Let $S(t) = \frac{1}{\pi}\Im \log\zeta\left(\frac{1}{2}+it\right)$. We prove
an unconditional lower bound on the measure of the sets $\{t\in [T,2T] \colon
S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log
T}\right)^{1/3}$. For $V \leq (\log T)^{1/3-\varepsilon}$ our bound has a
Gaussian shape with variance proportional to $\log\log T$. At the endpoint, $V
\asymp \left(\frac{\log T}{\log \log T}\right)^{1/3}$, our result implies the
best known $\Omega$-theorem for $S(t)$ which is due to Tsang. We also explain
how the method breaks down for $V \gg \left(\frac{\log T}{\log \log
T}\right)^{1/3}$ given our current knowledge about the zeros of the zeta
function. Conditionally on the Riemann hypothesis we extend our results to the
range $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log
T}\right)^{1/2}$.
