We make use of a 500 ks Chandra HRC-S/LETG spectrum of the blazar H 2356-309, combined with a lower signal-to-noise ratio (S/N; 100 ks) pilot LETG spectrum of the same target, to search for the presence of warm-hot absorbing gas associated with two large-scale structures (LSSs) crossed by this sight line and to constrain its physical state and geometry. Strong (log?N O VII ≥ 1016?cm–2) O VII Kα absorption associated with a third LSS crossed by this line of sight (the Sculptor Wall (SW)), at z = 0.03, has already been detected in a previous work. Here, we focus on two additional prominent filamentary LSSs along the same line of sight, one at z = 0.062 (the Pisces-Cetus Supercluster (PCS)) and another at z = 0.128 (the "Farther Sculptor Wall" (FSW)). The combined LETG spectrum has an S/N of ~11.6-12.6 per resolution element in the 20-25 ? and an average 3σ sensitivity to intervening O VII Kα absorption line equivalent widths (EWs) of EWO VII 14 m? in the available redshift range (z 0.165). No statistically significant (i.e., ≥3σ) individual absorption is detected from any of the strong He- or H-like transitions of C, O, and Ne (the most abundant metals in gas with solar-like composition) at the redshifts of the PCS and FSW structures and down to the EW thresholds mentioned above. However, we are still able to constrain the physical and geometrical parameters of the putative absorbing gas associated with these structures, by performing a joint spectral fit of various marginal detections and upper limits of the strongest expected lines with our self-consistent hybrid-ionization WHIM spectral model. At the redshift of the PCS, we identify a warm phase with log T = 5.35+0.07 –0.13 K and log N H = (19.1 ± 0.2) cm–2 possibly co-existing with a much hotter and statistically less significant phase with log T = 6.9+0.1 –0.8 K and log N H = 20.1+0.3 –1.7 cm–2 (1σ errors). These two separate physical phases are identified through, and mainly constrained by, C V Kα (warm phase) and O VIII Kα (hot phase) absorption, with single line significances of 1.5σ each. For the second LSS, at z 0.128, only one hot component is hinted in the data, through O VIII Kα (1.6σ) and Ne IX Kα (1.2σ). For this system, we estimate log T = 6.6+0.1 –0.2 K and log N H = 19.8+0.4 –0.8 cm–2. Our column density and temperature constraints on the warm-hot gaseous content of these two LSSs, combined with the measurements obtained for the hot gas permeating the SW, allow us to estimate the cumulative number density per unit redshifts of O VII WHIM absorbers at three different EW thresholds of 0.4 m?, 7 m?, and 25.8 m?. This is consistent with expectations only at the very low end of EW thresholds, but exceeds predictions at 7 m? and 25.8 m? (by more than 2σ). We also estimate the cosmological mass density of the WHIM based on the four absorbers we tentatively detect along this line of sight, obtaining ΩWHIM b = (0.021+0.031 –0.018)(Z/Z ☉)–1, consistent with the cosmological mass density of the intergalactic "missing baryons" only if we assume high metallicities (Z ~ Z ☉).
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