High efficiency, terawatt peak power X-ray Free Electron Lasers (XFELs) are a promising tool for enabling 3D atomic resolution single molecule imaging and nonlinear science using X-ray beams. Increasing the efficiency of XFELs while maintaining good longitudinal coherence can be achieved via self-seeding and tapering the undulator magnetic field. The efficiency of tapered self seeded XFELs is limited by two factors: the ratio of seed power to beam energy spread and the ratio of seed power to electron beam shot noise. We present a method to overcome these limitations by producing a strong X-ray seed and amplifying it with a small energy spread electron bunch. This can be achieved by selectively suppressing lasing for part of the electron beam in the SASE section and using the rest of the bunch to generate the seed radiation. In this manner one can reach saturation with the seeding electrons and the strong seed pulse can be overlapped with the “fresh" electrons downstream of the self-seeding monochromator. Simulations of this scenario demonstrating an increased efficiency are presented for two systems, an optimal superconducting undulator design and the Linac Coherent Light Source. In the case of the LCLS we examine how the betatron oscillations leading to selective suppression can be induced by using the transverse wakefield of a parallel plate corrugated structure, a dechirper.