  Bosonic super-WIMPs, including pseudoscalar and vector particles, are dark
matter candidates. Until now, many underground experiments searches for
super-WIMPs have been performed in the mass range of a few $\rm keV/c^2$ to 1
$\rm MeV/c^2$. All these searches utilize the absorption process of a
super-WIMP by a target atom in the detector, which is similar to the
photoelectric effect. In this study, we consider another process, namely, a
Compton-like process. As an example, we compare the cross-section of a
germanium atom for the absorption process with that of a Compton-like process.
Our findings indicate that the cross-section for the Compton-like process
becomes dominant, compared to that for the absorption process for mass above
approximately 150 $\rm keV/c^2$ for both pseudoscalar and vector super-WIMPs.
In particular, the cross-section for the Compton-like process for a vector
super-WIMP becomes increasingly greater than that for the absorption process by
an order of magnitude to two orders of magnitude in the 400 $\rm keV/c^2$ to 1
$\rm MeV/c^2$ mass range, respectively. By including the Compton-like process,
which has not been used in any other super-WIMP search experiment, the
experimental upper limits can be improved.
