Context. Vela X is a region of extended radio emission in the western part of the Vela constellation: one of the nearest pulsar wind nebulae, and associated with the energetic Vela pulsar (PSR B0833-45). Extended very-high-energy (VHE) γ-ray emission (HESS J0835-455) was discovered using the H.E.S.S. experiment in 2004. The VHE γ-ray emission was found to be coincident with a region of X-ray emission discovered with ROSAT above 1.5 keV (the so-called Vela X cocoon): a filamentary structure extending southwest from the pulsar to the centre of Vela X. Aims: A deeper observation of the entire Vela X nebula region, also including larger offsets from the cocoon, has been performed with H.E.S.S. This re-observation was carried out in order to probe the extent of the non-thermal emission from the Vela X region at TeV energies and to investigate its spectral properties. Methods: To increase the sensitivity to the faint γ-ray emission from the very extended Vela X region, a multivariate analysis method combining three complementary reconstruction techniques of Cherenkov-shower images is applied for the selection of γ-ray events. The analysis is performed with the On/Off background method, which estimates the background from separate observations pointing away from Vela X; towards regions free of γ-ray sources but with comparable observation conditions. Results: The γ-ray surface brightness over the large Vela X region reveals that the detection of non-thermal VHE γ-ray emission from the PWN HESS J0835-455 is statistically significant over a region of radius 1.2° around the position α = 08h35m00s, δ = -45°36'00'' (J2000). The Vela X region exhibits almost uniform γ-ray spectra over its full extent: the differential energy spectrum can be described by a power-law function with a hard spectral index Γ = 1.32 ± 0.06stat ± 0.12sys and an exponential cutoff at an energy of (14.0 ± 1.6stat ± 2.6sys) TeV. Compared to the previous H.E.S.S. observations of Vela X the new analysis confirms the general spatial overlap of the bulk of the VHE γ-ray emission with the X-ray cocoon, while its extent and morphology appear more consistent with the (more extended) radio emission, contradicting the simple correspondence between VHE γ-ray and X-ray emissions. Morphological and spectral results challenge the interpretation of the origin of γ-ray emission in the GeV and TeV ranges in the framework of current models.