Aims. We explore the high spectral resolution X-ray data towards the quasar 3C 273 to search for signals of hot (∼10⁶⁻⁷ K) X-ray-absorbing gas co-located with two established intergalactic far-ultraviolet (FUV) O VI absorbers. Methods. We analyze the soft X-ray band grating data of all XMM-Newton and Chandra instruments to search for the hot phase absorption lines at the FUV predicted redshifts. The viability of potential line detections is examined by adopting the constraints of a physically justified absorption model. The WHIM hypothesis is investigated with a complementary 3D galaxy distribution analysis and by detailed comparison of the measurement results to the WHIM properties in the EAGLE cosmological, hydrodynamical simulation. Results. At one of the examined FUV redshifts, z = 0.09017 ± 0.00003, we measured signals of two hot ion species, O VIII and Ne IX, with a 3.9σ combined significance level. While the absorption signal is only marginally detected in individual co-added spectra, considering the line features in all instruments collectively and assuming collisional equilibrium for absorbing gas, we were able to constrain the temperature (kT = 0.26 ± 0.03 keV) and the column density (N_H × Z_⊙/Z = 1.3_−0.5^+0.6 × 10¹⁹ cm⁻²) of the absorber. Thermal analysis indicates that FUV and X-ray absorption relate to different phases, with estimated temperatures, T_FUV ≈ 3 × 10⁵, and, T_{X − ray} ≈ 3 × 10⁶ K. These temperatures match the EAGLE predictions for WHIM at the FUV/X-ray measured N_ion-ranges. We detected a large scale galactic filament crossing the sight-line at the redshift of the absorption, linking the absorption to this structure. Conclusions. This study provides observational insights into co-existing warm and hot gas within a WHIM filament and estimates the ratio of the hot and warm phases. Because the hot phase is thermally distinct from the O VI gas, the estimated baryon content of the absorber is increased, conveying the promise of X-ray follow-up studies of FUV detected WHIM in refining the picture of the missing baryons.