By adding new gauge singlets of neutral leptons, the improved versions of the 3-3-1 models with right-handed neutrinos have been recently introduced in order to explain recent experimental neutrino oscillation data through the inverse seesaw mechanism. We prove that these models predict promising signals of lepton-flavor-violating decays of the standard-model-like Higgs boson $h_1^0\to \mu \tau ,e\tau$, which are suppressed in the original versions. One-loop contributions to these decay amplitudes are introduced in the unitary gauge. Based on a numerical investigation, we find that the branching ratios of the decays $h_1^0\to \mu \tau ,e\tau$ can reach values of ${10}^{-5}$ in the regions of parameter space satisfying the current experimental data of the decay $\mu \to e\gamma$. The value of ${10}^{-4}$ appears when the Yukawa couplings of leptons are close to the perturbative limit. Some interesting properties of these regions of parameter space are also discussed.