  The no-hair conjecture in General Relativity (GR) states that a Kerr black
hole (BH) is completely described by its mass and spin. As a consequence, the
complex quasi-normal-mode (QNM) frequencies of a binary-black-hole (BBH)
ringdown can be uniquely determined by the mass and spin of the remnant object.
Conversely, measurement of the QNM frequencies could be an independent test of
the no-hair conjecture. This paper extends to spinning BHs earlier work that
proposed to test the no-hair conjecture by measuring the complex QNM
frequencies of a BBH ringdown using parameterized inspiral-merger-ringdown
waveforms in the effective-one-body formalism, thereby taking full advantage of
the entire signal power and removing dependency on the predicted or estimated
start time of the ringdown. Our method was used to analyze the properties of
the merger remnants for BBHs observed by LIGO-Virgo in the first half of their
third observing (O3a) run. After testing our method with GR and non-GR
synthetic-signal injections in Gaussian noise, we analyze, for the first time,
two BBHs from the first (O1) and second (O2) LIGO-Virgo observing runs, and two
additional BBHs from the O3a run. We then provide joint constraints with
published results from the O3a run. In the most agnostic and conservative
scenario where we combine the information from different events using a
hierarchical approach, we obtain, at $90\%$ credibility, that the fractional
deviations in the frequency (damping time) of the dominant QNM are $\delta
f_{220}=0.03^{+0.10}_{-0.09}$ ($\delta \tau_{220}=0.10^{+0.44}_{-0.39}$),
respectively, an improvement of a factor of $\sim 4$ ($\sim 2$) over the
results obtained with our model in the LIGO-Virgo publication. The single-event
most-stringent constraint to date continues to be GW150914 for which we obtain
$\delta f_{220}=0.05^{+0.11}_{-0.07}$ and $\delta
\tau_{220}=0.07^{+0.26}_{-0.23}$.
