Context. In recent decades, thousands of substellar companions have been discovered with both indirect and direct methods of detection. While the majority of the sample is populated by objects discovered using radial velocity and transit techniques, an increasing number have been directly imaged. These planets and brown dwarfs are extraordinary sources of information that help in rounding out our understanding of planetary systems. Aims. In this paper, we focus our attention on substellar companions detected with the latter technique, with the primary goal of investigating their close surroundings and looking for additional companions and satellites, as well as disks and rings. Any such discovery would shed light on many unresolved questions, particularly with regard to their possible formation mechanisms. Methods. To reveal bound features of directly imaged companions, whether for point-like or extended sources, we need to suppress the contribution from the source itself. Therefore, we developed a method based on the negative fake companion technique that first estimates the position in the field of view (FoV) and the flux of the imaged companion with high precision, then subtracts a rescaled model point spread function (PSF) from the imaged companion, using either an image of the central star or another PSF in the FoV. Next it performs techniques, such as angular differential imaging, to further remove quasi-static patterns of the star (i.e., speckle contaminants) that affect the residuals of close-in companions. Results. After testing our tools on simulated companions and disks and on systems that were chosen ad hoc, we applied the method to the sample of substellar objects observed with SPHERE during the SHINE GTO survey. Among the 27 planets and brown dwarfs we analyzed, most objects did not show remarkable features, which was as expected, with the possible exception of a point source close to DH Tau B. This candidate companion was detected in four different SPHERE observations, with an estimated mass of ~1MJup, and a mass ratio with respect to the brown dwarf of 1∕10. This binary system, if confirmed, would be the first of its kind, opening up interesting questions for the formation mechanism, evolution, and frequency of such pairs. In order to address the latter, the residuals and contrasts reached for 25 companions in the sample of substellar objects observed with SPHERE were derived. If the DH Tau Bb companion is real, the binary fraction obtained is ~7%, which is in good agreement with the results obtained for field brown dwarfs. Conclusions. While there may currently be many limitations affecting the exploration of bound features to directly imaged exoplanets and brown dwarfs, next-generation instruments from the ground and space (i.e., JWST, ELT, and LUVOIR) will be able to image fainter objects and, thus, drive the application of this technique in upcoming searches for exo-moons and circumplanetary disks.