  We present numerically exact results for the spectral function of a single
charge carrier that is injected into an Ising chain at finite temperature $T$.
Both ferromagnetic and antiferromagnetic coupling between the Ising spins are
considered. The interaction between the carrier and the Ising spins is assumed
to be on-site and of Ising type, as well. We find that the carrier's spectral
function exhibits a distinctive fine structure of resonances that are due to
the temporary entrapping of the carrier inside small magnetic domains. The
connection to models of disordered binary alloys where similar effects can
occur is explained. We use these results to construct an accurate
(quasi)analytic approximation for low and medium-$T$ spectral functions.
