  In collisional thermometry, a system in contact with the thermal bath is
probed by a stream of ancillas. Coherences and collective measurements were
shown to improve the Fisher information in some parameter regimes, for a stream
of independent and identically prepared (i.i.d.) ancillas in some specific
states [Seah et al., Phys. Rev. Lett., 180602 (2019)]. Here we refine the
analysis of this metrological advantage by optimising over the possible input
ancilla states, also for block-i.i.d.~states of block size b=2. For both an
indirect measurement interaction and a coherent energy exchange channel, we
show when the thermal Cram\'er-Rao bound can be beaten, and when a collective
measurement of $N>1$ ancilla may return advantages over single-copy
measurements.
