  In probabilistic quantum metrology, one aims at finding weak measurements
that concentrate the Fisher Information on the resulting quantum states,
post-selected according to the weak outcomes. Though the Quantum Cram\'er-Rao
bound itself cannot be overshot this way, it could be possible to improve the
information-cost ratio, or even the total Fisher Information. We propose a
post-selection protocol achieving this goal based on single-photon subtraction
onto a thermal state of radiation yielding a greater information-cost ratio for
the temperature parameter with respect to the standard strategy required to
achieve the Quantum Cram\'er-Rao bound. We address just fully-classical states
of radiation: this contrasts with (but does not contradict) a recent result
proving that, concerning unitary quantum estimation problems, post-selection
strategies can outperform direct measurement protocols only if a particular
quasiprobability associated with the family of parameter-dependent quantum
states becomes negative, a clear signature of nonclassicality.
