A stable, single-input, single-output linear system is considered whose input is affected by biased multi-sinusoidal disturbances with uncertain frequencies belonging to a given range, unknown phases and amplitudes. The problem of exponential disturbance rejection by output feedback from any initial condition is addressed and solved. The knowledge of the sign of the real part (or the imaginary part) of the transfer functions, evaluated at the frequencies belonging to the given range, is sufficient to design a novel hybrid output feedback compensator. The critical parameters of the linear controller, which are related to the unknown frequencies, are updated every predetermined time interval and are generated by an exponentially converging adaptive observer, starting from any initial condition compatible with the given range. The output is exponentially driven to zero while the unknown disturbance is reconstructed.