Multi-stage harmonic cascade at seeded free-electron lasers

External seeded free-electron lasers (FELs) have exhibited substantial progress in diverse applications over the last decade. However, the frequency up-conversion efficiency in single-stage seeded FELs, particularly in high-gain harmonic generation (HGHG), remains constrained to a modest level. This limitation restricts its capability to conduct experiments within the 'water window'. This paper presents a novel method for generating coherent X-ray FEL pulses in the water window region based on the HGHG scheme with multi-stage harmonic cascade. Without any additional modifications to the HGHG configuration, simulation results demonstrate the generation of intense 3 nm coherent FEL radiation using an external ultraviolet seed laser. This indicates an increase of the harmonic conversion number to approximately 90. A preliminary experiment is performed to evaluate the feasibility of this method. The proposed approach could potentially serve as an efficient method to broaden the wavelength coverage accessible to both existing and planned seeded X-ray FEL facilities.