The computation of the amplitude, $\alpha$, 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, $\epsilon^2\ll1$, a new derivation of the ``beyond all orders in $\epsilon$'' amplitude, $\alpha$, is presented. It is shown by asymptotic matching techniques, extended to higher orders in $\epsilon$, that the value of $\alpha$ 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 [Proc. R. Soc. Lond. A 424, 19 (1989)], not only sheds new light on the computation of $\alpha$, but also greatly facilitates its numerical determination to a remarkable precision for so small values of $\epsilon$, which are beyond the capabilities of standard numerical methods.