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) gamma-ray emission (HESS J0835-455) was discovered using the H. E. S. S. experiment in 2004. The VHE gamma-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 gamma-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 gamma-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 gamma-ray sources but with comparable observation conditions. Results. The gamma-ray surface brightness over the large Vela X region reveals that the detection of non-thermal VHE gamma-ray emission from the PWN HESS J0835-455 is statistically significant over a region of radius 1.2 degrees around the position alpha = 08(h)35(m)00(s), delta = -45 degrees 36'00 '' (J2000). The Vela X region exhibits almost uniform gamma-ray spectra over its full extent: the differential energy spectrum can be described by a power-law function with a hard spectral index Gamma = 1.32 +/- 0.06(stat) +/- 0.12(sys) and an exponential cutoff at an energy of (14.0 +/- 1.6(stat) +/- 2.6(sys)) 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 gamma-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 gamma-ray and X-ray emissions. Morphological and spectral results challenge the interpretation of the origin of gamma-ray emission in the GeV and TeV ranges in the framework of current models.