Numerical and experimental studies have been performed to evaluate the enhancement of diffraction efficiency of diffraction gratings around B $K$-emission by overcoating lanthanum series layers on conventional metal-coated laminar-type gratings. We propose an optical design method based on the concept of spectral flux given by collection efficiency and diffraction efficiency. A diffraction grating with a small angle of incidence provides an advantage to soft x-ray spectrographs because it collects the emission at a larger solid angle compared to that of conventional grazing incidence diffraction gratings. Numerical calculations indicated that La and ${\rm{La}}{{\rm{F}}_3}$ were promising as overcoating materials on a laminar-type Ni-coated diffraction grating, and we performed an experimental study using ${\rm{La}}{{\rm{F}}_3}$ and La/C overcoatings, considering their producibility and durability. The diffraction efficiencies were measured using a reflectometer at a synchrotron facility. The diffraction efficiencies observed at 183.4 eV were 29.4% and 34.3% at angles of incidence of 85.1° and 84.9° for ${\rm{Ni}}/{\rm{La}}{{\rm{F}}_3}$ and Ni/La/C gratings, respectively.