Context. Despite tremendous progress in the detection and characterization of extrasolar planetary systems in the last 25 yr, we have not pinpointed any Solar System analogues. In particular, Jupiter-like planets (either mature or old) are barely detectable beyond 5 au with indirect techniques and they are still out of the reach of direct imaging techniques. Aims. Our study is aimed at a search for exoplanets throughout the whole ESO/VLT-SPHERE archive with an improved and unsupervised data analysis algorithm that could allow us to detect massive giant planets at 5 au. To prepare, test, and optimize our approach, we gathered a sample of 24 solar-type stars observed with SPHERE using angular and spectral differential imaging modes. Methods. We used PACO, a recently developed new-generation algorithm that has been shown to outperform classical methods. We also improved the SPHERE pre-reduction pipeline and optimized the outputs of PACO to enhance the detection performance. We developed custom-built spectral prior libraries to optimize the detection capability of the ASDI mode for both IRDIS and IFS. Results. Compared to previous works conducted with more classical algorithms, the contrast limits we derived with PACO are more reliable and significantly improved, especially at short angular separations, where a gain by a factor ten has been obtained between 0.2 and 0.5 arcsec. Under good observing conditions, planets down to 5 M Jup , orbiting at 5 au could be detected around stars within 60 parsec. We identified two exoplanet candidates that will require a follow-up to test for a common proper motion. Conclusions. In this work, we use a small sample to demonstrate the benefits of PACO in terms of achievable contrast and of control of the confidence levels. In addition, we have developed custom tools to take full advantage of this algorithm and to quantity the total error budget on the estimated astrometry and photometry. This work paves the way towards an end-to-end, homogeneous, and unsupervised massive re-reduction of archival direct imaging surveys in the quest for new exo-Jupiters.