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We study the rare decay process of $Z$ boson into photon, accompanied by aCP-even or CP-odd scalar. We present the analytical delineation of theprocesses through the model-independent parametrizations of the new physicscouplings and, finally, consider the Next-to-Minimal Supersymmetric StandardModel to mark out the parameter space where the branching fraction can have themaximum value. As a part of the necessary phenomenological and experimentalcross-checks, we aim to fit the anomalous magnetic moment of the muon and $W$boson mass anomaly through the supersymmetric contributions. We also find thatthe decays $Z\to H_1 \gamma, A_1 \gamma$ can serve as an excellentcomplementary test to $BR(h_{\rm SM}\to Z\gamma)$. In fact, to facilitatefuture searches, we unveil a few benchmark points that additionally satisfy thedeviation of $BR(h_{\rm SM}\to Z\gamma)$ from the SM value based on the recentmeasurements of ATLAS and CMS. Future proposals such as ILC, CEPC, and FCC-eeare anticipated to operate for multiple years, focusing on center-of-massenergy near the $Z$ pole. Consequently, these projects will be capable ofconducting experiments at the Giga-$Z$ ($10^{9}$ of $Z$ bosons) and Tera-$Z$($10^{12}$ of $Z$ bosons) phases, which may probe the aforesaid rare decayprocesses, thus the model as well. These unconventional yet complementarysearches offer different routes to explore the supersymmetric models withextended Higgs sectors like NMSSM.

Testing $Z$ boson rare decays $Z\to H_1 \gamma, A_1 \gamma$ with $(g-2)_{\mu}$, $\delta M_W$, and $BR(h_{\rm SM}\to Z\gamma)$ in the NMSSM