  Aims. To search for these rare objects, we study 32 Galactic O-type stars
that were reported as SB1s in the literature. In our sample we include Cyg X-1,
which is known to host an accreting stellar-mass BH, and HD 74194, a supergiant
fast X-ray transient, in order to validate our methodology. The final goal is
to characterise the nature of the unseen companions to determine if they are MS
stars, stripped helium stars, triples, or compact objects such as neutron stars
or stellar-mass BHs. Methods. After measuring radial velocities and deriving
orbital solutions for all the systems in our sample, we performed spectral
disentangling to extract putative signatures of faint secondary companions from
the composite spectra. We derived stellar parameters for the visible stars and
estimated the mass ranges of the secondary stars using the binary mass
function. Variability observed in the photometric TESS light curves was also
searched for indications of the presence of putative companions, degenerate or
not. Results. In 17 of the 32 systems reported as SB1s, we extract secondary
signatures, down to mass ratios of ~0.15. For the 17 newly detected
double-lined spectroscopic binaries (SB2s), we derive physical properties of
the individual components and discuss why they have not been detected as such
before. Among the remaining systems, we identify nine systems with possible NS
or low-mass MS companions. For Cyg X-1 and HD 130298, we are not able to
extract any signatures for the companions, and the minimum masses of their
companions are estimated to be about 7Msun. Our simulations show that
secondaries with such a mass should be detectable from our dataset, no matter
their nature: MS stars, stripped helium stars or even triples. While this is
expected for Cyg X-1, confirming our methodology, our simulations also strongly
suggest that HD 130298 could be another candidate to host a stellar-mass BH.
