As part of the ongoing CALIFA survey, we have conducted a thorough bidimensional analysis of the ionized gas in two E/S0 galaxies, NGC 6762 and NGC 5966, aiming to shed light on the nature of their warm ionized ISM. Specifically, we present optical (3745-7300 Å) integral field spectroscopy obtained with the PMAS/PPAK integral field spectrophotometer. Its wide field-of-view (1' × 1') covers the entire optical extent of each galaxy down to faint continuum surface brightnesses. To recover the nebular lines, we modeled and subtracted the underlying stellar continuum from the observed spectra using the STARLIGHT spectral synthesis code. The pure emission-line spectra were used to investigate the gas properties and determine the possible sources of ionization. We show the advantages of IFU data in interpreting the complex nature of the ionized gas in NGC 6762 and NGC 5966. In NGC 6762, the ionized gas and stellar emission display similar morphologies, while the emission line morphology is elongated in NGC 5966, spanning ~6 kpc, and is oriented roughly orthogonal to the major axis of the stellar continuum ellipsoid. Whereas gas and stars are kinematically aligned in NGC 6762, the gas is kinematically decoupled from the stars in NGC 5966. A decoupled rotating disk or an "ionization cone" are two possible interpretations of the elongated ionized gas structure in NGC 5966. The latter would be the first "ionization cone" of such a dimension detected within a weak emission-line galaxy. Both galaxies have weak emission-lines relative to the continuum[EW(Hα) ≲ 3 Å] and have very low excitation, log([Oiii]λ5007/Hβ) ≲ 0.5. Based on optical diagnostic ratios ([Oiii]λ5007/Hβ, [Nii]λ6584/Hα, [Sii]λ6717, 6731/Hα, [Oi]λ6300/Hα), both objects contain a LINER nucleus and an extended LINER-like gas emission. The emission line ratios do not vary significantly with radius or aperture, which indicates that the nebular properties are spatially homogeneous. The gas emission in NGC 6762 can be best explained by photoionization by pAGB stars without the need of invoking any other excitation mechanism. In the case of NGC 5966, the presence of a nuclear ionizing source seems to be required to shape the elongated gas emission feature in the "ionization cone" scenario, although ionization by pAGB stars cannot be ruled out. Further study of this object is needed to clarify the nature of its elongated gas structure. Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck-Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC).