  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.
