We present a detailed X-ray and UV high-time-resolution monitoring of the final reflaring phase of the 2022 outburst of the accreting millisecond pulsar SAX J1808.4−3658, based on simultaneous XMM-Newton and Hubble Space Telescope (HST) observations. The uninterrupted coverage provided by XMM-Newton enabled a detailed characterization of the spectral and temporal evolution of the source X-ray emission, as the flux varied by approximately one order of magnitude. We detected coherent X-ray pulsations during the whole X-ray observation, down to a 0.5–10 keV luminosity of L_{X(low), 0.5−10} ≃ 6.21_{0.20−0.15d²3.5 erg s⁻¹}; this is among the lowest ever observed in this source during the outburst state. At the lowest flux levels, we observed significant variations in pulse amplitude and phase. These variations were anticorrelated with the X-ray source flux. We found a sharp phase jump of ∼0.4 cycles, accompanied by a doubling of the pulse amplitude and a softening of the X-ray emission. We interpreted changes in the X-ray pulse profiles as drifts of emission regions on the neutron-star surface, driven by an increase in the inner-disk radius when the mass-accretion rate decreased. The dependence of the pulse phase on the X-ray flux was consistent with a magnetospheric radius scaling as R_m ∝ Ṁ^Λ, with Λ = −0.17(9), which is in broad agreement with theoretical predictions. Simultaneous HST observations confirmed the presence of significant UV pulsations at an X-ray luminosity approximately a factor of two lower than during the 2019 outburst, extending the range of mass accretion rates at which UV pulsations have been detected. The measured pulsed UV luminosity, L_pulsed^UV = 1.1(3) × 10³² erg s⁻¹, was consistent with that observed during the 2019 outburst. Yet, such a UV luminosity exceeds the predictions of standard emission models, as further confirmed by the shape of the pulsed spectral energy distribution.