  The P\"ochl-Teller effective potential mimics an asymptotically de Sitter
black hole bounded by an event horizon and a cosmological one. Owing to the
benefit of being analytically soluble, the asymptotic quasinormal modes in the
modified P\"oschl-Teller potential have been extensively explored in the
literature by various authors, and the results bear distinct features.
Specifically, for small discontinuities placed at the potential's peak, Skakala
and Visser showed that the resulting modes lie primarily along the imaginary
frequency axis, in line with the numerical results encountered for most black
hole metrics. However, it was also suggested that under ultraviolet
perturbations, asymptotic modes are expected to lie parallel to the real axis,
closely intervening with recent developments on spectral instability. In this
work, by numerical and semi-analytical approaches, we aim to resolve the above
apparent ambiguity. The numerical scheme is based on an improved version of the
matrix method, which is implemented in compactified hyperboloidal coordinates
on the Chebyshev grid. It is demonstrated that both asymptotic behaviors indeed
agree with the numerical findings, which is somewhat to one's surprise.
Specifically, we report the emergence of a novel branch of purely imaginary
modes originating from a bifurcation in the asymptotic quasinormal mode
spectrum. Moreover, we demonstrate how the bifurcation and asymptotic modes
evolve as the discontinuity moves away from the potential's peak, furnishing a
dynamic picture as the spectral instability unfolds. It is further argued that
they can be partly attributed to the observed parity-dependent deviations
occurring for the low-lying perturbed modes of the original P\"oschl-Teller
effective potential.
