Asymptotic flow states with limiting drag modification are explored via direct numerical simulations in a moderate-curvature viscoelastic Taylor–Couette flow of the FENE-P fluid. We show that asymptotic drag modification (ADM) states are achieved at different solvent-to-total viscosity ratios ($\beta$) by gradually increasing the Weissenberg number from 10 to 150. As $\beta$ decreases from 0.99 to 0.90, for the first time, a continuous transition pathway is realised from the maximum drag reduction to the maximum drag enhancement, revealing a complete phase diagram of the ADM states. This transition originates from the competition between Reynolds stress reduction and polymer stress development, namely, a mechanistic change in angular momentum transport. Reduced $\beta$ has been found to effectively enhance elastic instability, suppressing large-scale Taylor vortices while promoting the formation of small-scale elastic Görtler vortices. The enhancement and in turn dominance of small-scale structures result in stronger incoherent transport, facilitating efficient mixing and substantial polymer stress development that ultimately drives the AMD state transition. Further analysis of the scale-decomposed transport equation of turbulent kinetic energy reveals an inverse energy cascade in the gap centre, which is attributed to the polymer-induced energy redistribution: polymers extract more energy from large scales than they can dissipate, with the excess energy redirected to smaller scales. However, the energy accumulating at smaller scales cannot be dissipated immediately and is consequently transferred back to larger scales via nonlinear interactions, thereby unravelling a novel polymer-mediated cycle for the reverse energy cascade. Overall, this study unravels the challenging puzzle of the existence of distinct dynamically connected ADM states and paves the way for coordinated experimental, simulation and theoretical studies of transition pathways to desired ADM states.