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 $10^7leqtext{Ta}leq10^{11}$ are explored for both co- and counter-rotating configurations. In the $text{Ta}$ range $10^8leqtext{Ta}leq10^{10}$, 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.
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