High-irradiance short-pulse lasers incident on solid density thin foils provide high-energy, picosecond-duration, and monochromatic K(α) x-ray sources, but with limited conversion efficiency ϵ of laser energy into K(α) x-ray energy. A novel two-stage target concept is proposed that utilizes ultrahigh-contrast laser interactions with primary ultrathin foils in order to efficiently generate and transport in large quantities only the most effective K(α)-producing high-energy electrons into secondary x-ray converter foils. Benchmarked simulations with no free numerical parameters indicate an ϵ enhancement greater than tenfold over conventional single targets may be possible.