Antiferromagnetic materials promise improved performance for spintronic applications, as they are robust against external magnetic field perturbations and allow for faster magnetization dynamics compared to ferromagnets. The direct observation of the antiferromagnetic state, however, is challenging due to the absence of a macroscopic magnetization. Here, we show that the spin Hall magnetoresistance (SMR) is a versatile tool to probe the antiferromagnetic spin structure via simple electrical transport experiments by investigating the easy-plane antiferromagnetic insulators $\alpha$-Fe2O3 (hematite) and NiO in bilayer heterostructures with a Pt heavy metal top electrode. While rotating an external magnetic field in three orthogonal planes, we record the longitudinal and the transverse resistivities of Pt and observe characteristic resistivity modulations consistent with the SMR effect. We analyze both their amplitude and phase and compare the data to the results from a prototypical collinear ferrimagnetic Y3Fe5O12/Pt bilayer. The observed magnetic field dependence is explained in a comprehensive model, based on two magnetic sublattices and taking into account magnetic field-induced modifications of the domain structure. Our results show that the SMR allows us to understand the spin configuration and to investigate magnetoelastic effects in antiferromagnetic multi-domain materials. Furthermore, in $\alpha$-Fe2O3/Pt bilayers, we find an unexpectedly large SMR amplitude of $2.5 \times 10^{-3}$, twice as high as for prototype Y3Fe5O12/Pt bilayers, making the system particularly interesting for room-temperature antiferromagnetic spintronic applications.