  We reformulate in Hamiltonian language the recent proposal by Hohm and
Zwiebach of an action yielding the most general $O(d,d)$-symmetric string
cosmology equations, at tree-level in the string-loop expansion, but to all
orders in the $\alpha'$ expansion. This allows us to give a simple
characterization of a large class of non-singular, non-perturbative, pre-big
bang scenarios smoothly interpolating between a low-energy initial accelerated
(string frame) expansion and a phase of final (string and Einstein frame)
decelerated expansion. Interestingly, these solutions must necessarily include,
just around the bounce, a very short phase of (string-frame) contraction.
