Measurement of the transverse polarization of $\Lambda^0$ ($\bar\Lambda^0$) in the $e^+e^-$ continuum at centre-of-mass energies near 10.58 GeV. The polarization is measured as a function of the momentum fraction and transverse momentum. The polarization is also measured for events with a $\pi^\pm$ or $K^\pm$ in the opposite hemisphere as a function of the momentum fractions of the $\Lambda^0$ and charged meson.The polarization as a function of the momentum fraction for $\Lambda^0$ baryons produced in $(uds)$ events, and for prompt $\Lambda^0$ and those from the decay of $\Sigma^0$ mesons is also measured.