We present a new suite of late-end reionization simulations performed with ATON-HE, a revised version of the GPU-based radiative transfer code ATON that includes helium. The simulations are able to reproduce the Ly$\alpha$ flux distribution of the E-XQR-30 sample of QSO absorption spectra at $5 \lesssim z \lesssim 6.2$, and show that a large variety of reionization models are consistent with these data. We explore a range of variations in source models and in the early-stage evolution of reionization. Our fiducial reionization history has a midpoint of reionization at $z = 6.5$, but we also explore an `Early' reionization history with a midpoint at $z = 7.5$ and an `Extremely Early' reionization history with a midpoint at $z = 9.5$. Haloes massive enough to host observed Ly$\alpha$ emitters are highly biased. The fraction of such haloes embedded in ionized bubbles that are large enough to allow high Ly$\alpha$ transmission becomes close to unity much before the volume filling factor of ionized regions. For our fiducial reionization history this happens at $z = 8$, probably too late to be consistent with the detection by JWST of abundant Ly$\alpha$ emission out to $z = 11$. A reionization history in our `Early' model or perhaps even our `Extremely Early' model may be required, suggesting a Thomson scattering optical depth in tension with that reported by Planck, but consistent with recent suggestions of a significantly higher value.