Context. S stars are late-type giants with overabundances of s-process elements. They come in two flavors depending on the presence or lack of presence of technetium (Tc), an element without stable isotopes. Intrinsic S stars are Tc-rich and genuine asymptotic giant branch (AGB) stars, while extrinsic S stars owe their s-process over abundances to the pollution from a former AGB companion, which is now a white dwarf (WD). In addition to Tc, another distinctive feature between intrinsic and extrinsic S stars is the overabundance of niobium (Nb) in the latter class. Indeed, since the mass transfer occurred long ago, 93Zr had time to decay into the only stable isotope of Nb, 93Nb, causing its overabundance. Aims. We discuss the case of the S stars BD+79°156 and o1 Ori, whose specificity lies in sharing the distinctive features of both intrinsic and extrinsic S stars, namely the presence of Tc along with a Nb overabundance. Methods. We used high-resolution HERMES optical spectra, MARCS model atmospheres of S stars, Gaia DR2 parallaxes, and STAREVOL evolutionary tracks to determine the stellar parameters and chemical abundances of the two S stars, and to locate them in the Hertzsprung-Russell (HR) diagram. Results. BD+79°156 is the first clear case of a bitrinsic star, that is, a doubly s-process-enriched object, first through mass transfer in a binary system and then through internal nucleosynthesis that is responsible for the Tc-enrichment in BD+79°156, which must, therefore, have reached the AGB phase of its evolution. This hybrid nature of the s-process pattern in BD+79°156 is supported by its binary nature and its location in the HR diagram that is just beyond the onset of the third dredge-up on the AGB. The Tc-rich, binary S-star o1 Ori with a WD companion was another long-standing candidate for a similar hybrid s-process enrichment. However, the marginal overabundance of Nb derived in o1 Ori does not allow one to trace evidence of large amounts of pollution coming from the AGB progenitor of its current WD companion unambiguously. As a side product, the current study offers a new way of detecting binary AGB stars with WD companions by identifying their Tc-rich nature along with a Nb overabundance.