  In the small dispersion limit, solutions to the Korteweg-de Vries equation
develop an interval of fast oscillations after a certain time. We obtain a
universal asymptotic expansion for the Korteweg-de Vries solution near the
leading edge of the oscillatory zone up to second order corrections. This
expansion involves the Hastings-McLeod solution of the Painlev\'e II equation.
We prove our results using the Riemann-Hilbert approach.
