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The Be X-ray pulsar (BeXRP) SXP 59 underwent a giant outburst in 2017 with a peak X-ray luminosity of 1.1 × 1038 erg s−1. We report on the X-ray behaviour of SXP 59 with the XMM–Newton and NuSTAR observations collected at the outburst peak, decay, and the low luminosity states. The pulse profiles are energy dependent, the pulse fraction increases with the photon energy and saturates at 65 per cent above 10 keV. It is difficult to constrain the change in the geometry of emitting region with the limited data. Nevertheless, because the pulse shape generally has a double-peaked profile at high luminosity and a single peak profile at low luminosity, we prefer the scenario that the source transited from the super-critical state to the sub-critical regime. This result would further imply that the neutron star (NS) in SXP 59 has a typical magnetic field. We confirm that the soft excess revealed below 2 keV is dominated by a cool thermal component. On the other hand, the NuSTAR spectra can be described as a combination of the non-thermal component from the accretion column, a hot blackbody emission, and an iron emission line. The temperature of the hot thermal component decreases with time, while its size remains constant (R ∼ 0.6 km). The existence of the hot blackbody at high luminosity cannot be explained with the present accretion theories for BeXRPs. It means that either more sophisticated spectral models are required to describe the X-ray spectra of luminous BeXRPs, or there is non-dipole magnetic field close to the NS surface. [ABSTRACT FROM AUTHOR]