Context. The architecture of planetary systems hosting ultra-short-period (USP) planets is a key diagnostic for understanding formation and migration scenarios. The presence of outer giant companions in these systems is of particular interest to test theories regarding dynamical effects and pebble accretion. Aims. We present an extended radial velocity (RV) monitoring of the bright star K2-312≡HD 80653, known to host a rocky USP super-Earth ( P _b = 0.720 d). Previous studies identified a long-term trend and subsequently a Keplerian signal due to an outer highly eccentric giant planet, K2-312 c. We aim to refine the orbital parameters of K2-312 c by precisely monitoring its periastron passage and to model the formation and dynamical evolution of the system. Methods. We analyzed a set of 237 HARPS-N high-resolution spectra, extending the observation baseline of previous literature by almost 4 years. We performed a joint analysis of the RVs together with K2 and TESS photometry to refine the ephemerides and properties of the two planets. To account for stellar activity, we coupled the Keplerian models with a Gaussian processes regression. Results. K2-312 c is a cold Jupiter on a wide orbit (orbital period refined to P _c = 871.32 d), with a minimum mass of M _c sin i ~5 M _Jup and a refined eccentricity of e _c ~ 0.85. It is among the most eccentric cold Jupiters known in multi-planet systems, and the only one that is highly eccentric and has a USP planet companion. Our simulations suggest that planet-planet scattering between two giant planets could have driven K2-312 c to its current high eccentricity, ejected the other giant, and still allowed for the survival of K2-312 b. The extended observation baseline further allowed us to identify the stellar rotation period and a long activity cycle, while a new K2 reduction improved the significance of the secondary eclipse detection for K2-312 b.

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