  Using the Riemann-Hilbert approach, we explicitly construct the asymptotic
$\Psi$-function corresponding to the solution $y\sim\pm\sqrt{-x/2}$ as
$|x|\to\infty$ to the second Painlev\'e equation $y_{xx}=2y^3+xy-\alpha$. We
precisely describe the exponentially small jump in the dominant solution and
the coefficient asymptotics in its power-like expansion.
