  Context. B supergiants (BSGs) represent an important connection between the
main sequence and more extreme evolutionary stages of massive stars.
Additionally, lying toward the cool end of the hot star regime, determining
their wind properties is crucial to constrain the evolution and feedback of
massive stars as, for instance, they might manifest the bi-stability jump
phenomenon. Aims. We undertake a detailed analysis of a representative sample
of 18 Small Magellanic Cloud (SMC) BSGs within the ULLYSES and XShootU
datasets. Our UV and optical analysis spans BSGs from B0 to B8 - covering the
bi-stability jump region. We aim to evaluate their evolutionary status and
verify what their wind properties say about the bi-stability jump in a
low-metallicity environment. Methods. We used the CMFGEN to model the spectra
and photometry (from UV to infrared) of our sample. We compare our results with
different evolutionary models, with previous determinations in the literature
of OB stars, and with diverging mass-loss recipes at the bi-stability jump.
Additionally, we provide the first BSG models in the SMC including X-rays.
Results. (i) Within a single-stellar evolution framework, the evolutionary
status of early BSGs seem less clear than that of late BSGs, which agree with
H-shell burning models. (ii) UV analysis shows evidence that BSGs contain
X-rays in their atmospheres, for which we provide constraints. In general, we
find higher X-ray luminosity (close to the standard log(L_X/L) ~ -7) for early
BSGs. For cooler BSGs, lower values are preferred, log(L_X/L) ~ -8.5. (iii) The
obtained mass-loss rates suggest neither a jump nor a monotonic decrease with
temperature. Instead, a rather constant trend is observed, which is at odds
with the increase found for Galactic BSGs. (iv) The wind velocity behavior with
temperature shows a sharp drop at ~19 kK, similar to what is observed for
Galactic BSGs.
