Future searches for bio-markers on habitable exoplanets will rely on telescope instruments that achieve extremely high contrast at small planet-to-star angular separations. Coronagraphy is a promising starlight suppression technique, providing excellent contrast and throughput for off-axis sources on clear apertures. However, the complexity of space- and ground-based telescope apertures goes on increasing over time, owing to the combination of primary mirror segmentation, the secondary mirror, and its support structures. These discontinuities in the telescope aperture limit the coronagraph performance. In this paper, we present ACAD-OSM, a novel active method to correct for the diffractive effects of aperture discontinuities in the final image plane of a coronagraph. Active methods use one or several deformable mirrors that are controlled with an interaction matrix to correct for the aberrations in the pupil. However, they are often limited by the amount of aberrations introduced by aperture discontinuities. This algorithm relies on the recalibration of the interaction matrix during the correction process to overcome this limitation. We first describe the ACAD-OSM technique and compare it to the previous active methods for the correction of aperture discontinuities. We then show its performance in terms of contrast and off-axis throughput for static aperture discontinuities (segmentation, struts) and for some aberrations evolving over the life of the instrument (residual phase aberrations, artifacts in the aperture, misalignments in the coronagraph design). This technique can now obtain the Earth-like planet detection threshold of contrast on any given aperture over at least a 10% spectral bandwidth, with several coronagraph designs.