We explore scenarios for reionizing the intergalactic medium with low galaxy ionizing photon escape fractions. We combine simulation-based halo mass–dependent escape fractions with an extrapolation of the observed galaxy rest-ultraviolet luminosity functions to solve for the reionization history from z = 20\to 4. We explore the posterior distributions for key unknown quantities, including the limiting halo mass for star formation, the ionizing photon production efficiency, and a potential contribution from active galactic nuclei (AGNs). We marginalize over the allowable parameter space using a Markov chain Monte Carlo method, finding a solution that satisfies the most model-independent constraints on reionization. Our fiducial model can match observational constraints with an average escape fraction of <5% throughout the bulk of the epoch of reionization if (i) galaxies form stars down to the atomic cooling limit before reionization and a photosuppression mass of log(M h/M ⊙) ∼ 9 during/after reionization (‑13 < M UV,lim <‑11), (ii) galaxies become more efficient producers of ionizing photons at higher redshifts and fainter magnitudes, and (iii) there is a significant but subdominant contribution by AGNs at z ≲ 7. In this model, the faintest galaxies (M UV >‑15) dominate the ionizing emissivity, leading to an earlier start to reionization and a smoother evolution of the ionized volume-filling fraction than models that assume a single escape fraction at all redshifts and luminosities. The ionizing emissivity from this model is consistent with observations at z = 4–5 (and below, when extrapolated), in contrast to some models that assume a single escape fraction. Our predicted ionized volume-filling fraction at z = 7 of {Q}{{{H}}II}} = 78% (±8%) is in modest (∼1σ–2σ) tension with observations of Lyα emitters at z ∼ 7 and the damping-wing analyses of the two known z > 7 quasars, which prefer {Q}{{{H}}II},z=7} ∼ 40%–50%.