We investigate geometric percolation in two-dimensional assemblies of randomly positioned, widthless sticks of uniform length. The stick orientations are sampled from probability density functions with a vanishing mean orientational order parameter, while their higher-order structure varies continuously from perfect isotropy to perfect cross-alignment. Monte Carlo simulations show that the percolation threshold is lowest for the isotropic distribution and increases toward the perfectly cross-aligned limit. This variation is strongly nonuniform: small orientational perturbations near perfect cross-alignment produce substantially larger changes in the threshold than comparable perturbations near isotropy. The simulation results are interpreted using an orientation-resolved eigenvalue formulation based on the full excluded-area kernel and a fully preaveraged excluded-area approximation. For the probability density functions considered, the two formulations predict nearly identical thresholds. Orientation-dependent variations in the expected contact numbers therefore provide only a minor correction, whereas the dominant orientational dependence is retained by the averaged excluded area between pairs of sticks. A harmonic representation of this quantity further shows that the second moment captures the leading variation near isotropy, while progressively higher-order moments are required as the distribution approaches perfect cross-alignment. The results identify the orientational information that must be retained to predict geometric percolation in distributions having the same mean order parameter but different higher-order structures.