  We use experiments and direct numerical simulations to probe the phase-space
of low-curvature Taylor--Couette (TC) flow in the vicinity of the ultimate
regime. The cylinder radius ratio is fixed at $\eta=r_i/r_o=0.91$.
Non-dimensional shear drivings (Taylor numbers $\text{Ta}$) in the range
$10^7\leq\text{Ta}\leq10^{11}$ are explored for both co- and counter-rotating
configurations. In the $\text{Ta}$ range $10^8\leq\text{Ta}\leq10^{10}$, we
observe two local maxima of the angular momentum transport as a function of the
cylinder rotation ratio, which can be described as either as "co-" and
"counter-rotating" due to their location or as "broad" and "narrow" due to
their shape. We confirm that the broad peak is accompanied by the strengthening
of the large-scale structures, and that the narrow peak appears once the
driving (Ta) is strong enough. As first evidenced in numerical simulations by
Brauckmann \emph{et al.}~(2016), the broad peak is produced by centrifugal
instabilities and that the narrow peak is a consequence of shear instabilities.
We describe how the peaks change with $\text{Ta}$ as the flow becomes more
turbulent. Close to the transition to the ultimate regime when the boundary
layers (BLs) become turbulent, the usual structure of counter-rotating Taylor
vortex pairs breaks down and stable unpaired rolls appear locally. We attribute
this state to changes in the underlying roll characteristics during the
transition to the ultimate regime. Further changes in the flow structure around
$\text{Ta}\approx10^{10}$ cause the broad peak to disappear completely and the
narrow peak to move. This second transition is caused when the regions inside
the BLs which are locally smooth regions disappear and the whole boundary layer
becomes active.
