  We combine the ideas of a Harish-Chandra--Howe local character expansion,
which can be centred at an arbitrary semisimple element, and a Kim--Murnaghan
asymptotic expansion, which so far has been considered only around the
identity. We show that, for most smooth, irreducible representations (those
containing a good, minimal K-type), Kim--Murnaghan-type asymptotic expansions
are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple
elements. We then give an explicit, inductive recipe for computing the
coefficients in an asymptotic expansion for a tame supercuspidal
representation. The only additional information needed in the inductive step is
a fourth root of unity, which we expect to be useful in proving stability and
endoscopic-transfer identities.
