During the epoch of reionization (EoR), the 21-cm signal allows direct observation of the neutral hydrogen (HI) in the intergalactic medium (IGM). In the post-reionization era, this signal instead probes HI in galaxies, which traces the dark matter density distribution. With new numerical simulations, we investigated the end stages of reionization to elucidate the transition of our Universe into the post-reionization era. Our models are consistent with the latest high-redshift measurements, including ultraviolet (UV) luminosity functions up to redshift similar or equal to 8. Notably, these models consistently reproduced the evolution of the UV photon background, which is constrained from Lyman-alpha absorption spectra. We studied the dependence of this background on the nature of photon sinks in the IGM, requiring mean free path of UV photons to be similar to 10 comoving-megaparsecs (cMpc) during the EoR that increases gradually with time during late stages (z less than or similar to 6). Our models revealed that the reionization of the IGM transitioned from an inside-out to an outside-in process when the Universe is less than 0.01 per cent neutral. During this epoch, the 21-cm signal also shifted from probing predominantly the H I in the IGM to that in galaxies. Furthermore, we identified a statistically significant number of large neutral islands (with sizes up to 40 cMpc) persisting until very late stages (5 less than or similar to z less than or similar to 6) that can imprint features in Lyman-alpha absorption spectra and also produce a knee-like feature in the 21-cm power spectrum.