We examine the temperature structure of the intergalactic medium (IGM) surrounding a hard radiation source, such as a quasi-stellar object (QSO), as it responds to the onset of helium reionization by the source. We model the reionization using a radiative transfer (RT) code coupled to a particle-mesh (PM) N-body code. Neutral hydrogen and helium are initially ionized by a starburst spectrum, which is allowed to gradually evolve into a power law spectrum (∝ ν−0.5). Multiple simulations were performed with different times for the onset and dominance of the hard spectrum, with onset redshifts ranging from z= 3.5 to 5.5. The source is placed in a high-density region to mimic the expected local environment of a QSO. Simulations with the source placed in a low-density environment were also performed as control cases to explore the role of the environment on the properties of the surrounding IGM. We find in both cases that the IGM temperature within the He iii region produced exceeds the IGM temperature before full helium reionization, resulting in a ‘thermal proximity effect’, but that the temperature in the He iii region increases systematically with distance from the source. With time the temperature relaxes with a reduced spread as a function of impact parameter along neighbouring lines of sight, although the trend continues to persist until z= 2. Such a trend could be detected using the widths of intervening metal absorption systems using high resolution, high signal-to-noise ratio spectra. By contrast, the Doppler widths of H i absorption features in mock spectra along neighbouring lines of sight show a weak trend with impact parameter prior to full helium reionization reflecting the behaviour of the underlying density and peculiar velocity fields, but they take on a near constant value after helium reionization, with a median value near 25 km s−1 at z= 3, in good agreement with observations.