The stability of mass transfer in binaries with convective giant donors remains an open question in modern astrophysics. There is a significant discrepancy between what the existing methods predict for a response to mass loss of the giant itself, as
well as for the mass transfer rate during the Roche lobe overflow. Here we show that the recombination energy in the superadiabatic layer plays an important and hitherto unaccounted-for role in he donors response to mass loss, in particular on its luminosity and effective temperature. Our improved optically thick nozzle method to calculate the mass transfer rate via $L_1$ allows us to evolve binary systems for a substantial Roche lobe overflow. We propose a new, strengthened criterion for the mass transfer instability, basing it on whether the donor experiences overflow through its outer Lagrangian point. We find that with the new criterion, if the donor has a well-developed outer convective envelope, the critical initial mass ratio for which a binary would evolve stably through the conservative mass transfer varies from $1.5$ to $2.2$, which is about twice as large as previously believed. In underdeveloped giants with shallow convective envelopes this critical ratio may be even larger. When the convective envelope is still growing, and in particular for most cases of massive donors, the critical mass ratio gradually decreases to this value, from that of radiative donors.
