We calculate the contribution to the polarization of $Lambda$ hyperons in relativistic nuclear collisions at high energy from the decays of $Sigma^*(1385)$ and $Sigma^0$, which are the predominant sources of $Lambda$ production besides the primary component, as a function of the $Lambda$ momentum. Particularly, we estimate the longitudinal component of the mean spin vector as a function of the azimuthal angle in the transverse plane, assuming that primary $Sigma^*$ and $Sigma^0$ polarization follow the predictions of local thermodynamic equilibrium in a relativistic fluid. Provided that the rapidity dependence around midrapidity of polarization is negligible, we find that this component of the overall spin vector has a very similar pattern to the primary one. Therefore, we conclude that the secondary decays cannot account for the discrepancy in sign between experimental data and hydrodynamic model predictions of the longitudinal polarization of $Lambda$ hyperons recently measured by the STAR experiment at RHIC.
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