In late 2008, the quasi-persistent neutron star X-ray transient and eclipsing binary EXO 0748−676 started a transition from outburst to quiescence, after it actively accreted for more than 24 yr. In a previous work, we discussed Chandra and Swift observations obtained during the first 5 months of this transition. Here, we report on further X-ray observations of EXO 0748−676, extending the quiescent monitoring to 1.6 yr. Chandra and XMM–Newton data reveal quiescent X-ray spectra composed of a soft, thermal component that is well fitted by a neutron star atmosphere model. An additional hard power-law tail is detected that changes non-monotonically over time, contributing between 4 and 20 per cent to the total unabsorbed 0.5–10 keV flux. The combined set of Chandra, XMM–Newton and Swift data reveals that the thermal bolometric luminosity fades from ∼ 1 × 1034 to 6 × 1033 (D/7.4 kpc)2 erg s −1, whereas the inferred neutron star effective temperature decreases from ∼124 to 109 eV. We interpret the observed decay as cooling of the neutron star crust and show that the fractional quiescent temperature change of EXO 0748−676 is markedly smaller than observed for three other neutron star X-ray binaries that underwent prolonged accretion outbursts.