  The attenuation of Ly$\alpha$ photons by neutral hydrogen in the
intergalactic medium (IGM) at $z\gtrsim5$ continues to be a powerful probe for
studying the epoch of reionisation. Given a framework to estimate the intrinsic
(true) Ly$\alpha$ emission of high-$z$ sources, one can infer the ionisation
state of the IGM during reionisation. In this work, we use the enlarged XQR-30
sample of 42 high-resolution and high-SNR QSO spectra between
$5.8\lesssim\,z\lesssim\,6.6$ obtained with VLT/X-Shooter to place constraints
on the IGM neutral fraction. This is achieved using our existing Bayesian QSO
reconstruction framework which accounts for uncertainties such as the: (i)
posterior distribution of predicted intrinsic Ly$\alpha$ emission profiles
(obtained via covariance matrix reconstruction of the Ly$\alpha$ and N V
emission lines from unattenuated high-ionisation emission line profiles; C IV,
Si IV + O IV] and C III]) and (ii) distribution of ionised regions within the
IGM using synthetic damping wing profiles drawn from a $1.6^3$ Gpc$^3$
reionisation simulation. Following careful quality control, we used 23 of the
42 available QSOs to obtain constraints/limits on the IGM neutral fraction
during the tail-end of reionisation. Our median and 68th percentile constraints
on the IGM neutral fraction are: $0.20\substack{+0.14\\-0.12}$ and
$0.29\substack{+0.14\\-0.13}$ at $z = 6.15$~and 6.35. Further, we also report
68th percentile upper-limits of $\bar{x}_{\mathrm{H\,{\scriptscriptstyle I}}} <
0.21$, 0.20, 0.21 and 0.18 at $z = 5.8, 5.95, 6.05$~and 6.55. These results
imply reionisation is still ongoing at $5.8\lesssim\,z\lesssim\,6.55$,
consistent with previous results from XQR-30 (dark fraction and Ly$\alpha$
forest) along with other observational probes considered in the literature.
