  We propose to combine the nonlinear scaling fields associated with the
high-temperature (HT) fixed point, with those associated with the unstable
fixed point, in order to calculate the susceptibility and other thermodynamic
quantities. The general strategy relies on simple linear relations between the
HT scaling fields and the thermodynamic quantities, and the estimation of RG
invariants formed out of the two sets of scaling fields. This estimation
requires convergent expansions in overlapping domains. If, in addition, the
initial values of the scaling fields associated with the unstable fixed point
can be calculated from the temperature and the parameters appearing in the
microscopic Hamiltonian, one can estimate the critical amplitudes. This
strategy has been developed using Dyson's hierarchical model where all the
steps can be approximately implemented with good accuracy. We show numerically
that for this model (and a simplified version of it), the required overlap
apparently occurs, allowing an accurate determination of the critical
amplitudes.
