  A perturbative scheme is applied to calculate corrections to the leading,
exponentially small (beyond-all-orders) amplitude of the ``trailing'' wave
asymptotics of weakly localized solitons. The model considered is a Korteweg-de
Vries equation modified by a fifth order derivative term,
$\epsilon^2\partial_x^5$ with $\epsilon\ll1$ (fKdV). The leading order
corrections to the tail amplitude are calculated up to ${\cal{O}}(\epsilon^5)$.
An arbitrary precision numerical code is implemented to solve the fKdV equation
and to check the perturbative results. Excellent agreement is found between the
numerical and analytical results. Our work also clarifies the origin of a
long-standing disagreement between the ${\cal{O}}(\epsilon^2)$ perturbative
result of Grimshaw and Joshi [SIAM J. Appl. Math. 55, 124 (1995)] and the
numerical results of Boyd [Comp. Phys. 9, 324 (1995)].
