Measurements of the ionisation state of the intergalactic medium (IGM) can probe the sources of the extragalactic ionising background. We provide new measurements of the ionising emissivity of galaxies using measurements of the ionising background and ionising photon mean free path from high-redshift quasar spectra at 2.5 < z < 6. Unlike most prior works, we account for radiative-transfer effects and possible neutral islands from the tail of reionisation at z > 5. We combine our results with measurements of the UV luminosity function to constrain the average escaping ionising efficiency of galaxies, . Assuming galaxies with MUV < –11 emit ionizing photons, we find , and 1σ upper limits of 24.48 and 24.31 at z = 2.5 and 4, respectively. We also estimate the population-averaged fesc using measurements of intrinsic ionising efficiency from JWST. We find and 6, and 1σ upper limits of fesc < 0.138 and 0.096 at z = 2.5 and 4, respectively, for MUV < –11. Our findings are consistent with prior measurements of , but indicate a factor of several increase between z = 4 and 6. The steepness of this evolution is sensitive to the highly uncertain mean free path and ionising background intensity at z > 5. Lastly, we find photons per H atom are emitted into the IGM between z = 6 and = 5.3. This is ≈ 4× more than needed to complete the last 20% of reionisation absent recombinations, suggesting that reionisation’s end was likely absorption-dominated.