  This work studies the tail exponents for the height function of the
stationary stochastic six vertex model in the moderate deviations regime. For
the upper tail of the height function we find upper and lower bounds of
matching order, with a tail exponent of $\frac{3}{2}$, characteristic of KPZ
distributions. We also obtain an upper bound for the lower tail of the same
order.
  Our results for the stochastic six vertex model hold under a restriction on
the model parameters for which a certain "microscopic concavity" condition
holds. Nevertheless, our estimates are sufficiently strong to pass through the
degeneration of the stochastic six vertex model to the ASEP. We therefore
obtain tail estimates for both the current as well as the location of a second
class particle in the ASEP with stationary (Bernoulli) initial data. Our
estimates complement the variance bounds obtained in the seminal work of
Bal\'azs and Sepp\"al\"ainen.}
