The recent measurement of an ionising mean free path $lambda_{text{mfp}}<1$ pMpc at $z=6$ challenges our understanding of the small-scale structure of the intergalactic medium (IGM) at the end of reionisation. We introduce a new method to constrain mfp at $z=6$ by using lower limits on the individual free paths of ionisation around quasars. Lyman-limit absorbers with a density sufficient to halt ionising photons produce strong absorption in the 6 lowest-energy Lyman transitions, in the absence of which a robust lower limit can be placed on the individual free path. Applying this method to a set of $26$ quasars at $5.5<z<6.5$, we find that $80%$ of bright quasars ($M_{1450}<-26.5$) require individual free paths larger than $2$ pMpc. We model the relation between opacity $kappa$ and photo-ionisation rate $Gamma$ via the parameter $xi$ such that $kappaproptoGamma^{-xi}$, and pose joint limits on mfp and $xi$. For the nominal value of $xi=2/3$, we constrain $lambda_{text{mfp}}>0.31 (0.18)$ pMpc at $2sigma (3sigma)$: a much tighter lower bound than obtained through traditional stacking methods. Our constraints get significantly stronger for lower values of $xi$. New constraints on mfp and $xi$ are crucial to our understanding of the reionisation-era IGM.
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