  We show that a sufficient condition for a subset $E$ in the Heisenberg group
(endowed with the Carnot-Carath\'{e}odory metric) to be contained in a
rectifiable curve is that it satisfies a modified analogue of Peter Jones's
geometric lemma. Our estimates improve on those of \cite{FFP}, by replacing the
power $2$ of the Jones-$\beta$-number with any power $r<4$. This complements
(in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we
showed that such an estimate was necessary, but with $r=4$.
