The computation of the amplitude, α, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg–de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term, ε2≪1, a new derivation of the “beyond all orders in ε” amplitude, α, is presented. It is shown by asymptotic matching techniques, extended to higher orders in ε, that the value of α can be obtained from the asymmetry at the center of the unique solution exponentially decaying in one direction. This observation, complemented by some fundamental results of Hammersley and Mazzarino [], not only sheds new light on the computation of α but also greatly facilitates its numerical determination to a remarkable precision for so small values of ε, which are beyond the capabilities of standard numerical methods. Published by the American Physical Society 2024