About 50 years ago, it was discovered that $\Lambda^0$ hyperons are produced polarized in collisions of unpolarized protons on beryllium. Despite enormous experimental and theoretical efforts, the origin of this polarization remains inconclusive to date. The $\Lambda^0$ polarization has also been observed in various collision systems, from $e^+$+$e^-$ to heavy-ion collisions. A recently proposed technique for the investigation of the $\Lambda^0$ hyperon polarization is a measurement of $\Lambda^0\bar{\Lambda}^0$, $\Lambda^0\Lambda^0$, and $\bar{\Lambda}^0\bar{\Lambda}^0$ spin-spin correlations. This technique is expected to help understand if the polarization is generated at early stages of the collisions, e.g. from initial state parton spin correlation, or if it is a final state effect originating from hadronization.

In these proceedings, we present a status of the first measurement utilizing this new experimental method in $p$+$p$ collisions at $\sqrt{s} = 200\,\textrm{GeV}$ by the STAR experiment. The $\Lambda^0$ and $\bar{\Lambda}^0$ candidates are reconstructed at mid-rapidity ($|y| < 1$) and in two transverse momentum ($p_\mathrm{T}$) bins which allows us to extract the $\Lambda^0$-hyperon spin-spin correlations for various $p_\mathrm{T}$ combinations of hyperons in $\Lambda^0\bar{\Lambda}^0$, $\Lambda^0\Lambda^0$, and $\bar{\Lambda}^0\bar{\Lambda}^0$ pairs. This measurement will provide new insight into $\Lambda^0$ hyperon spin polarization in $p$+$p$ collisions at RHIC energies.