We analyze the lepton flavor-violating (LFV) Higgs decay h -> tau mu in three supersymmetric models: the minimal supersymmetric Standard Model, supersymmetric seesaw model, and supersymmetric B-L model with inverse seesaw (BLSSM-IS). We show that it is not possible to enhance BR(h ->tau mu) without violating the experimental bound on the BRd (tau -> mu gamma) in the generic minimal supersymmetric Standard Model, with nonuniversal slepton masses and/or trilinear couplings. In the supersymmetric seesaw model, where flavor mixing is radiatively generated, the LFV process mu -> e gamma strongly constrains the parameter space, and the maximum value of BR(h -> tau mu) is of order 10(-10), which is much smaller than the recent results reported by the CMS and ATLAS experiments. On the other hand, in the BLSSM-IS with universal soft supersymmetry-breaking terms defined at the grand unified scale, BR(h -> tau mu) can be accommodated in a wide region of the parameter space without violating LFV constraints. Even though the background processes in the LHC have very small cross sections, we also analyze the possible signal within the BLSSM-IS over the relevant background processes and show that a signal can be extracted. The analysis shows that confirmation of the LFV Higgs boson decay reported by CMS and ATLAS would favor the BLSSM-IS class of models.