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In this paper we reanalyze the issue of fine-tuning in supersymmetric modelswhich feature Generalized Gauge Mediation (GGM) in the light of recentmeasurement of the mass of the light Higgs particle and taking into accountavailable data on the value of the muon magnetic moment $g_\mu-2$. We considerGGM models with 3, 5 and 6 input parameters and reduce the fine-tuning byassuming simple relations between them at the high scale. We are able to findsolutions which give the correct value of the light Higgs mass and are lessfine-tuned than models with standard gauge mediation (and with gravitymediation), however one never finds fine-tung measure lower than about $10^2$if one neglects the data on $g_\mu-2$ and and about four times more if onetakes the constraint given by $g_\mu-2$ into account. In general the current$g_\mu-2$ data push the models towards the high fine-tuning region. It isinteresting to note, that once one removes the contributions to the finetuninginduced by $\mu$ and $B_\mu$, then in the case with neglected $g_\mu-2$constraint one can easily find realistic vacua with fine-tuning of order 1 orlower, while the fine-tung remains always large when the $g_\mu-2$ constraintis enforced. One should note, that in the last case even a small shift of thelight Higgs mass towards smaller values both reduces fine-tuning and helps toimprove agreement of a model with $g_\mu-2$ data.

Fine-tuning in GGM and the 126 GeV Higgs particle