Nonlinear responses in transport experiments may unveil information and generate new phenomena in materials that are not accessible at linear order due to symmetry constraints. While the linear anomalous Hall response strictly requires the absence of time-reversal symmetry, the second-order, thus nonlinear, Hall response needs broken inversion symmetry. Recently, much effort has been made to obtain a second-order Hall voltage in response to a longitudinal ac driving current, both to obtain information about band geometric quantities and for its useful technological applications, including rectification and frequency doubling. Typically, additional material engineering is required in noncentrosymmetric systems to obtain second-order responses since it obeys a stringent crystallographic symmetry constraint. To circumvent this, an alternative route is to apply a dc electric field. In this Letter, we uncover an electric field induced second-order anomalous Hall effect in inversion-broken systems possessing experimentally accessible unconventional Rashba bands. We establish that the quantum metric, a geometrical feature of electronic wave functions providing information on the nontrivial structure of Bloch bands, is responsible for providing the nonlinear Hall response.