  In this paper we consider wildly ramified power series, \emph{i.e.}, power
series defined over a field of positive characteristic, fixing the origin,
where it is tangent to the identity. In this setting we introduce a new
invariant under change of coordinates called the \emph{second residue fixed
point index}, and provide a closed formula for it. As the name suggests this
invariant is closely related to the residue fixed point index, and they
coincide in the case that the power series have small multiplicity. Finally, we
characterize power series with large multiplicity having the smallest possible
multiplicity at the origin under iteration, in terms of this new invariant.
