We present the results of our abundance analysis of Fe, Ni, and O in high signal-to-noise ratio, high-resolution Very Large Telescope UVES and McDonald 2dcoude spectra of nine dwarfs and three giants in the Hyades open cluster. The difference in Fe abundances derived from Fe II and Fe I lines ([Fe /H] - [Fe /H]) and Ni I abundances derived from moderately high-excitation (χ ≈ 4.20 eV) lines is found to increase with decreasing Teff for the dwarfs. Both of these findings are in concordance with previous results of overexcitation/overionization in cool young dwarfs. Oxygen abundances are derived from the [O I] λ6300 line, with careful attention given to the Ni I blend. The dwarf O abundances are in star-to-star agreement within uncertainties, but the abundances of the three coolest dwarfs (4573 K ≤ Teff ≤ 4834 K) evince an increase with decreasing Teff. Possible causes for the apparent trend are considered, including the effects of overdissociation of O-containing molecules. O abundances are derived from the near-UV OH λ3167 line in high-quality Keck HIRES spectra, and no such effects are found; indeed, the OH-based abundances show an increase with decreasing Teff, leaving the nature and reality of the cool dwarf [O I]-based O trend uncertain. The mean relative O abundance of the six warmest dwarfs (5075 K ≤ Teff ≤ 5978 K) is [O/H] = +0.14 ± 0.02, and we find a mean abundance of [O/H] = +0.08 ± 0.02 for the giants. Thus, our updated analysis of the [O I] λ6300 line does not confirm the Hyades giant-dwarf oxygen discrepancy initially reported by King & Hiltgen, suggesting that the discrepancy was a consequence of analysis-related systematic errors. LTE oxygen abundances from the near-IR, high-excitation O I triplet are also derived for the giants, and the resulting abundances are approximately 0.28 dex higher than those derived from the [O I] line, in agreement with non-LTE predictions. Non-LTE corrections from the literature are applied to the giant triplet abundances; the resulting mean abundance is [O/H] = +0.17 ± 0.02, in decent concordance with the giant and dwarf [O I] abundances. Finally, Hyades giant and dwarf O abundances derived from the [O I] λ6300 line and high-excitation triplet, as well as dwarf O abundances derived from the near-UV OH λ3167 line, are compared, and a mean cluster O abundance of [O/H] = +0.12 ± 0.02 is achieved, which represents the best estimate of the Hyades O abundance.