Context. Exoplanet detections and characterizations via direct imaging require high contrast and high angular resolution. These requirements are typically pursued by combining (i) cutting-edge instrumental facilities equipped with extreme adaptive optics and coronagraphic systems, (ii) optimized differential imaging to introduce a diversity between the signals of the sought-for objects and that of the star, and (iii) dedicated (post-)processing algorithms to further eliminate the residual stellar leakages. Aims. With respect to the third technique, substantial efforts have been undertaken over this last decade on the design of more efficient post-processing algorithms. The whole data collection and retrieval processes currently allow to detect massive exoplanets at angular separations greater than a few tenths of au. The performance remains upper-bounded at shorter angular separations due to the lack of diversity induced by the processing of each epoch of observations individually. We aim to propose a new algorithm that is able to combine several observations of the same star by accounting for the Keplerian orbital motion across epochs of the sought-for exoplanets in order to constructively co-add their weak signals. Methods. The proposed algorithm, PACOME , integrates an exploration of the plausible orbits of the sought-for objects within an end-to-end statistical detection and estimation formalism. The latter is extended to a multi-epoch combination of the maximum likelihood framework of PACO , which is a post-processing algorithm of single-epoch observations. From this, we derived a reliable multi-epoch detection criterion, interpretable both in terms of probability of detection and of false alarm. In addition, PACOME is able to produce a few plausible estimates of the orbital elements of the detected sources and provide their local error bars. Results. We tested the proposed algorithm on several datasets obtained from the VLT/SPHERE instrument with IRDIS and IFS using the pupil tracking mode of the telescope. By resorting to injections of synthetic exoplanets, we show that PACOME is able to detect sources remaining undetectable by the most advanced post-processing of each individual epoch. The gain in detection sensitivity scales as high as the square root of the number of epochs. We also applied PACOME on a set of observations from the HR 8799 star hosting four known exoplanets, which can be detected by our algorithm with very high signal-to-noise ratios. Conclusions. PACOME is an algorithm for combining multi-epoch high-contrast observations of a given star. Its sensitivity and the reliability of its astrophysical outputs permits the detection of new candidate companions at a statistically grounded confidence level. In addition, its implementation is efficient, fast, and fully automatized.