  The high temperature expansion (HTE) of the specific heat of a spin system
fails at low temperatures, even if it is combined with a Pad\'e approximation.
On the other hand we often have information about the low temperature
asymptotics (LTA) of the system. Interpolation methods combine both kind of
information, HTE and LTA, in order to obtain an approximation of the specific
heat that holds for the whole temperature range. Here we revisit the entropy
method that has been previously published and propose two variants that better
cope with problems of the entropy method for gapped systems. We compare all
three methods applied to the antiferromagnetic Haldane spin-one chain and
especially apply the second variant, called Log Z method, to the cuboctahedron
for different spin quantum numbers. In particular, we demonstrate that the
interpolation method is able to detect an extra low-temperature maximum in the
specific heat that may appear if a separation of two energy scales is present
in the considered system. Finally we illustrate how interpolation also works
for classical spin systems.
