Nanometer-scale magnetization configurations known as magnetic skyrmions have mostly been studied in cubic chiral helimagnets, in which they are Bloch-type and their axes align along the applied magnetic field. In contrast, the orientation of Néel-type skyrmions is locked to the polar axis of the host material's underlying crystal structure. In the lacunar spinels GaV4S8 and GaV4Se8, the Néel-type skyrmion lattice phase exists for externally applied magnetic fields parallel to this axis and withstands oblique magnetic fields up to some critical angle. Here, we map out the stability of the skyrmion lattice phase in both crystals as a function of field angle and magnitude using dynamic cantilever magnetometry. The measured phase diagrams reproduce the major features predicted by a recent theoretical model, including a reentrant cycloidal phase in GaV4Se8. Nonetheless, we observe a greater robustness of the skyrmion phase to oblique fields, suggesting possible refinements to the model. Besides identifying transitions between the cycloidal, skyrmion lattice, and ferromagnetic states in the bulk, we measure additional anomalies in GaV4Se8 and assign them to magnetic states confined to polar structural domain walls.