There have been numerous attempts, both experimentally and theoretically, to understand the origin of the unexpectedly large transverse single-spin asymmetries ($A_N$) for inclusive hadron production at forward rapidity in p$^\uparrow$+p collisions that persist from low to high center-of-mass energies. Two proposed potential sources are the twist-3 contributions in the collinear factorization and the transverse-momentum-dependent contributions from either the initial-state quark and gluon Sivers functions or the final-state Collins fragmentation function. To investigate the underlying physics leading to this large $A_N$, we study $\pi^0$ $A_N$ with different topologies -- isolated and non-isolated, and $A_N$ for electromagnetic jets (EM-jets) of different substructures using Forward Meson Spectrometer (FMS) detector at STAR. Jet $A_N$ is sensitive to the initial state effect and can provide access to Sivers functions. To investigate final-state effects, we measure the Collins asymmetry of $\pi^0$ inside EM-jets. We present the most recent results for these asymmetries from p$^\uparrow$+p collisions at 200 GeV and 500 GeV. We also present new preliminary results of $A_N$ for EM-jets in FMS and Endcap Electromagnetic Calorimeter (EEMC) using p$^\uparrow$+p collisions at 200 GeV where we explore the dependences of $A_N$ on photon multiplicity inside the jet, jet transverse momentum, and jet energy. These results provide rich information towards understanding the physics mechanism of large $A_N$ in hadron collisions.