We investigate the behavior of a Josephson junction consisting of a ferromagnetic insulator-superconductor (FI-S) bilayer tunnel coupled to a superconducting electrode. We show that the Josephson coupling in the structure is strengthened by the presence of the spin-splitting field induced in the FI-S bilayer. Such strengthening manifests itself as an increase of the critical current I c with the amplitude of the exchange field. Furthermore, the effect can be strongly enhanced if the junction is taken out of equilibrium by a temperature bias. We propose a realistic setup to assess experimentally the magnitude of the induced exchange field, and predict a drastic deviation of the I c (T ) curve (T is the temperature) with respect to equilibrium. The interplay between superconductivity and ferromagnetism in superconductor-ferromagnet (S-F) hybrids exhibits a large variety of effects studied along the last years [1,2]. Experimental research mainly focuses on the control of the 0-π transition in the S-F-S junctions [3,4] (S-F-S) and on the creation, detection, and manipulation of triplet correlations in S-F hybrids [5][6][7][8][9]. From a fundamental point of view, the key phenomenon for the understanding of these effects is the proximity effect in S-F hybrids, and how the interplay between superconducting and magnetic correlations affect their thermodynamic and transport properties.While most of the theoretical and experimental investigations on S-F structures deal mainly with the penetration of the superconducting order into the ferromagnetic regions, it is also widely known that magnetic correlations can be induced in the superconductor via the inverse proximity effect [10][11][12][13]. If the ferromagnet is an insulator (FI), on the one hand, superconducting correlations are weakly suppressed at the FI-S interface and a finite exchange field, with an amplitude smaller than the superconducting gap 0 , is induced at the interface. Such exchange correlations penetrate into the bulk of S over distances of the order of the coherence length [10]. This results in a splitting of the density of states (DOS) of the superconductor, as observed in a number of experiments [14][15][16][17]. Yet, the spin-split DOS of a superconductor may lead to interesting effects such as, for instance, the absolute spinvalve effect [18][19][20], the magnetothermal Josephson valve [21,22], and the large enhancement of the Josephson coupling observed in F-S-I-S-F junctions (I stands for a conventional insulator) when the magnetic configuration of the F layers is arranged in the antiparallel state [23][24][25][26].In this paper, we show that an enhancement of the Josephson effect between two tunnel-coupled superconductors S L and S R can also be achieved if a unique FI is attached to one of the S electrodes, for instance, the left lead, as shown schematically in Fig. 1(a). According to the discussion above, the presence of the FI splits the DOS in the left superconductor. In principle, the presence of the spin-splitting field causes * sebastian_...