The triply differential cross section of unpolarized bremsstrahlung produced by polarized electrons in the Coulomb field of a nucleus is calculated with the aid of Sommerfeld-Maue wave functions. The photon asymmetry originating from transversely polarized electrons is calculated for various values of energies and angles, in particular for the parameters of recent experiments. As a function of the atomic number Z of the target nucleus, the asymmetry shows a distinct deviation from a linear dependence, in contrast to a previous calculation exact to lowest order in the Coulomb parameter Z/137. The cross-section formula is integrated numerically over the angles of the outgoing electrons to obtain the photon asymmetry observed in non-coincidence experiments.