  We consider a quantum lattice spin model featuring exact quasiparticle towers
of eigenstates with low entanglement at finite size, known as quantum many-body
scars (QMBS). We show that the states in the neighboring part of the energy
spectrum can be superposed to construct entire families of low-entanglement
states whose energy variance decreases asymptotically to zero as the lattice
size is increased. As a consequence, they have a relaxation time that diverges
in the thermodynamic limit, and therefore exhibit the typical behavior of exact
QMBS although they are not exact eigenstates of the Hamiltonian for any finite
size. We refer to such states as \textit{asymptotic} QMBS. These states are
orthogonal to any exact QMBS at any finite size, and their existence shows that
the presence of an exact QMBS leaves important signatures of non-thermalness in
the rest of the spectrum; therefore, QMBS-like phenomena can hide in what is
typically considered the thermal part of the spectrum. We support our study
using numerical simulations in the spin-1 XY model, a paradigmatic model for
QMBS, and we conclude by presenting a weak perturbation of the model that
destroys the exact QMBS while keeping the asymptotic QMBS.
