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58Ni +64Ni is the first case where the influence of positive Q-value transfer channels on sub-barrier fusion was evidenced, in a very well known experiment by Beckerman et al., by comparing with the two systems 58Ni + 58Ni and 64Ni+64Ni. Subsequent measurements on 64Ni + 64Ni showed that fusion hindrance is clearly present in this case. On the other hand, no indication of hindrance can be observed for 58Ni + 64Ni down to the measured level of 0.1 mb. In the present experiment the excitation function has been extended by two orders of magnitude downward. The cross sections for 58Ni + 64Ni continue decreasing very smoothly below the barrier, down to '1 µb. The logarithmic slope of the excitation function increases slowly, showing a tendency to saturate at the lowest energies. No maximum of the astrophysical S -factor is observed. Coupled-channels (CC) calculations using a Woods-Saxon potential and includinginelastic excitations only, underestimate the sub-barrier cross sections by a large amount. Good agreement is found by adding two-neutron transfer couplings to a schematical level. This behaviour is quite different from what already observed for 64Ni+ 64Ni (no positive Q-value transfer channels available), where a clear low-energy maximum of the S -factorappears, and whose excitation function is overestimated by a standard Woods-Saxon CC calculation. No hindrance effect is observed in 58Ni+ 64Ni in the measured energy range. This trend at deep sub-barrier energies reinforces the recent suggestion that the availability of several states following transfer with Q>0, effectively counterbalances the Pauli repulsion that, in general, is predicted to reduce tunneling probability inside the Coulomb barrier.