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The chirp-taper free-electron laser (FEL) scheme has been proposed to generate subfemtosecond broadband pulses using laser-modulated electron bunches in x-ray FELs. In this paper, we extend the chirp-taper concept to the postsaturation superradiant regime by combining an energy-modulated beam with a sinelike undulator taper. We show that subfemtosecond duration pulses with peak powers approaching a terawatt may be obtained with typical x-ray FEL parameters such as those at the Linac Coherent Light Source. The method shown here can produce a stable source of attosecond-scale, high-power x-ray pulses suitable for probing electronic dynamics within molecules.