We present studies of single-spin asymmetries for neutral pion electroproduction in semi-inclusive deep-inelastic scattering of 5.776 GeV polarized electrons from an unpolarized hydrogen target, using the CEBAF Large Acceptance Spectrometer (CLAS) at the Thomas Jefferson National Accelerator Facility. A substantial $\sin \phi_h$ amplitude has been measured in the distribution of the cross section asymmetry as a function of the azimuthal angle $\phi_h$ of the produced neutral pion. The dependence of this amplitude on Bjorken $x$ and on the pion transverse momentum is extracted with significantly higher precision than previous data and is compared to model calculations.