This study explored the influence of foil thickness, laser pulse width, and laser intensity to optimize the multi-keV X-ray conversion efficiency of a sandwiched (CH/Sn/CH) planar target under laser irradiation at the Shenguang II laser facility. The X-ray photon field values were measured using a set of elliptically bent crystal spectrometers and the conversion efficiencies (ξx) of photon energies were in the range of 3.7-4.3 keV. The experimental results indicate that the X-ray yields of 3.7 to 4.3 keV radiation strongly depend on the laser pulse width, target thickness, and laser intensity. The results also demonstrate that three-layer thin foils can provide an efficient multi-keV X-ray source because they can change the distribution of emitted multi-keV X-rays and target dynamics versus nanosecond laser pulses to produce large, hot, and underdense plasma. However, the underdense plasma produced as a rarefaction wave causes the overdense plasma generated by the laser pulse to expand. Therefore, the laser parameters and foil thickness must be carefully optimized to produce an efficient 3.7 to 4.3 keV X-ray source. Otherwise, the rarefaction waves from both sides of the thin foil may suppress multi-keV X-ray emission. This study represents an important advancement in the development of an efficient multi-keV L-shell-band X-ray source.