fits have indicated a non-zero value for the lepton mixing angle $\theta_{13}^l$ for some time [@firsttheta13nonzero]. Direct evidence for a large $\theta^l_{13}$ has been provided in 2011 by T2K [@T2K], MINOS [@MINOS] and Double Chooz [@DC]; for global fits including the results of T2K and MINOS see [@fogli; @schwetz; @maltoni]. Recently, Daya Bay [@DayaBay] have published their first result \^2 2 \^l\_[13]{}=0.0920.016 0.005 , while RENO [@RENO] find \^2 2 \^l\_[13]{}=0.1130.013 0.019 , implying $\theta^l_{13}\approx 0.15 \div 0.17$. Many models predicting TB mixing [@HPS] at LO, in particular those with the flavour symmetries $A_4$ and $S_4$, for reviews see [@reviews], are severely challenged by such a large value of $\theta_{13}^l$, because subleading corrections are too small to explain $\theta^l_{13}\sim 0.15$.

In this note we revisit an existing SUSY $S_4 \times SU(5)$ model [@HKL10]. In its original form, it predicts TB mixing in the neutrino sector, and the dominant source of the mixing angle $\theta_{13}^l$ is the (12)-rotation in the charged lepton sector which in turn is related to the Cabibbo angle $\theta_C$ so that $\theta^l_{13} \approx \theta_C/(3 \sqrt{2}) \approx 0.05$. We show that a minimal extension of this model, namely adding one flavon $\eta$, being a singlet under $S_4$, naturally 