After the successful determination of the reactor neutrino mixing angle ��_13 ~ 0.16 \neq 0, a new feature suggested by the current neutrino oscillation data is a sizeable deviation of the atmospheric neutrino mixing angle ��_23 from ��/4. Using the fact that the neutrino mixing matrix U = U_e^\dagger U_��, where U_e and U_��result from the diagonalisation of the charged lepton and neutrino mass matrices, and assuming that U_��has a i) bimaximal (BM), ii) tri-bimaximal (TBM) form, or else iii) corresponds to the conservation of the lepton charge L' = L_e - L_��- L_��(LC), we investigate quantitatively what are the minimal forms of U_e, in terms of angles and phases it contains, that can provide the requisite corrections to U_��so that ��_13, ��_23 and the solar neutrino mixing angle ��_12 have values compatible with the current data. Two possible orderings of the 12 and the 23 rotations in U_e, "standard" and "inverse", are considered. The results we obtain depend strongly on the type of ordering. In the case of "standard" ordering, in particular, the Dirac CP violation phase ��, present in U, is predicted to have a value in a narrow interval around i) ��~ ��in the BM (or LC) case, ii) ��~ 3��/2 or ��/2 in the TBM case, the CP conserving values ��= 0, ��, 2��being excluded in the TBM case at more than 4��. In the addendum we discuss the implications of the latest 2013 data.

35 pages, 10 figures. Version 3: addendum with an update of the results based on global analyses including the 2013 data (pages 31-35)