  Excited state contamination remains one of the most challenging sources of
systematic uncertainty to control in lattice QCD calculations of nucleon matrix
elements and form factors: early time separations are contaminated by excited
states and late times suffer from an exponentially bad signal-to-noise problem.
High-statistics calculations at large time separations $\gtrsim1$ fm are
commonly used to combat these issues. In this work, focusing on $g_A$, we
explore the alternative strategy of utilizing a large number of relatively
low-statistics calculations at short to medium time separations (0.2--1 fm),
combined with a multi-state analysis. On an ensemble with a pion mass of
approximately 310 MeV and a lattice spacing of approximately 0.09 fm, we find
this provides a more robust and economical method of quantifying and
controlling the excited state systematic uncertainty. A quantitative separation
of various types of excited states enables the identification of the transition
matrix elements as the dominant contamination. The excited state contamination
of the Feynman-Hellmann correlation function is found to reduce to the 1% level
at approximately 1 fm while for the more standard three-point functions, this
does not occur until after 2 fm. Critical to our findings is the use of a
global minimization, rather than fixing the spectrum from the two-point
functions and using them as input to the three-point analysis. We find that the
ground state parameters determined in such a global analysis are stable against
variations in the excited state model, the number of excited states, and the
truncation of early-time or late-time numerical data.
