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Barium (Ba), CH, and extrinsic or Tc-poor S-type stars are evolved low- and intermediate-mass stars that show enhancement of slow-neutron-capture-process elements on their surface, an indication of mass accretion from a former asymptotic giant branch (AGB) companion, which is now a white dwarf (WD). Ba and CH stars can be found in the main-sequence (MS), the sub-giant, and the giant phase, while extrinsic S-type stars populate the giant branches only. As these polluted stars evolve, they might be involved in a second phase of interaction with their now white dwarf companion. In this paper, we consider systems composed of a main-sequence Ba star and a WD companion when the former evolves along the Red Giant Branch (RGB). We want to determine if the orbital properties of the known population of Ba, CH, and S giants can be inferred from the evolution of their suspected dwarf progenitors. For this purpose, we use the BINSTAR binary evolution code and model MS+WD binary systems, considering different binary interaction mechanisms, such as a tidally-enhanced wind mass-loss and a reduced circularisation efficiency. To explore their impact on the second RGB ascent, we compare the modelled orbits with the observed period and eccentricity distributions of Ba and related giants. We show that, independently of the considered mechanism, there is a strong period cut off below which core-He burning stars should not be found in binary systems with a WD companion. This limit is shorter for more massive RGB stars and for more metal-poor systems. However, we still find a few low-mass short-period giant systems that are difficult to explain with our models as well as two systems with very high eccentricities.