  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
$h^0_1\rightarrow \mu\tau,e\tau$, 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 $h^0_1\rightarrow\mu\tau,e\tau$ can reach values
of $10^{-5}$ in the regions of parameter space satisfying the current
experimental data of the decay $\mu\rightarrow e\gamma$. 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.
