The archetypal model of the buckling of a compressed long elastic strut resting on a nonlinear elastic foundation is studied. Localised buckling is investigated when the foundation has both quadratic and cubic terms which initially destabilise but subsequently restabilise the structure. The primary solution can be detected by a double-scale perturbation procedure and is reminiscent of a solitary wave: essentially, it consists of a fast periodic oscillation which is slowly modulated and decays exponentially in both directions. Particular interest is paid to the process of adapting the procedure to account for the post-buckling behaviour of two-packet or double-humped solitary waves in this model. We employ the methods of beyond-all-orders asymptotics to reveal terms formally exponentially small in the perturbation parameter which have macroscopic effects on the post-buckling behaviour of the system including the interaction phenomenon of interest. The analysis is reinforced by direct numerical computations which reveal the so-called snaking behaviour in the subsidiary homoclinic orbits as is observed in the case of primary solutions. However, additional phenomena arise for these subsidiary forms including the formation of bridges linking solution paths and the appearance of a multitude of closed isolated loops disconnected from other features of the bifurcation diagram.