  We study the multiple scattering Sunyaev-Zeldovich (SZ) signature, extending
our previous analysis to high-temperature clusters. We consistently treat the
anisotropy of the ambient radiation field caused by the first scattering and
also consider lowest order kinematic terms. We show that due to temperature
corrections monopole through octupole anisotropy of the singly scattered SZ
signal attain different spectra in the second scattering. The difference
becomes more pronounced at high temperature, and thus could be used to
constrain individual line of sight moments of the electron density and
temperature profiles. While very challenging from the observational point of
view, this further extends the list of possible SZ observables that will be
important for 3D cluster-profile reconstruction, possibly helping to break
geometric degeneracies caused by projection effects. We also briefly discuss
the scattering of primordial CMB anisotropies by SZ clusters.
