Caterina Ballocco, Alessandro Papitto, Arianna Miraval Zanon, Giulia Illiano, Tiziana Di Salvo, Filippo Ambrosino, Luciano Burderi, Sergio Campana, Francesco Coti Zelati, Alessandro Di Marco, Christian Malacaria, Maura Pilia, Juri Poutanen, Tuomo Salmi, Andrea Sanna

We present a detailed X-ray/UV high-time resolution monitoring of the final reflaring phase of the accreting millisecond pulsar SAX~J1808.4$-$3658. During its 2022 outburst, we obtained simultaneous XMM-Newton and Hubble Space Telescope (HST) observations. We detected coherent X-ray pulsations down to a 0.5-10~keV luminosity of $L_{X(low)\,0.5-10} \simeq 6.21^{+0.20}_{-0.15} \times 10^{34} \, d^2_{3.5}\,\mathrm{erg \, s^{-1}}$, among the lowest ever observed in this source. The uninterrupted coverage provided by XMM-Newton enabled a detailed characterisation of the spectral and temporal evolution of the source X-ray emission as the flux varied by approximately one order of magnitude. At the lowest flux levels, we observed significant variations in pulse amplitude and phase correlated with the X-ray source flux. We found a sharp phase jump of $\sim 0.4$ cycles, accompanied by a doubling of the pulse amplitude and a softening of the X-ray emission. We interpret the changes in the X-ray pulse profiles as drifts of emission regions on the neutron star surface due to an increase of the inner disk radius occurring when the mass accretion rate decreases. The phase evolution was consistent with a magnetospheric radius scaling as $R_{m} \propto \dot{M}^{\Lambda}$, with $\Lambda = -0.17(9)$, in broad agreement with theoretical predictions. Simultaneous HST observations confirmed the presence of significant UV pulsations. The measured pulsed luminosity $-$ $L_{pulsed}^{UV} = (9 \pm 2) \times 10^{31} \, \text{erg} \, \text{s}^{-1}$ $-$ was approximately half that observed during the 2019 outburst, but the pulsed X-ray to UV luminosity ratio simultaneously measured remained consistent. Yet, such a UV luminosity exceeds the predictions of standard emission models, as further confirmed by the shape of the pulsed spectral energy distribution.