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Many theories beyond the Standard Model postulate short-range modifications to gravity which produce deviations of Newton's gravitational potential from a strict $1/r$ dependence. It is common to analyze experiments searching for these modifications using a potential of the form $V^{\prime}(r)=-\frac{GMm}{r} [1+\alpha \exp{(-r/\lambda)}]$. The best present constraints on $\alpha$ for $\lambda <100$\,nm come from neutron scattering and often employ comparisons of different measurements of the coherent neutron scattering amplitudes $b$. We analyze the internal consistency of existing data from two different types of measurements of low energy neutron scattering amplitudes: neutron interferometry, which involves squared momentum transfers $q^{2}=0$, and neutron gravity reflectometry, which involves squared momentum transfers $q^{2}=8mV_{opt}$ where $m$ is the neutron mass and $V_{opt}$ is the neutron optical potential of the medium. We show that the fractional difference $\frac{\Delta b}{|b|}$ averaged over the 7 elements where high precision data exists on the same material from both measurement methods is $[2.2 \pm 1.4] \times 10^{-4}$. We also show that $\frac{\Delta b}{|b|}$ for this data is insensitive both to exotic Yukawa interactions and also to the electromagnetic neutron-atom interactions proportional to the neutron-electron scattering length $b_{ne}$ and the neutron polarizability scattering amplitude $b_{pol}$. This result will be useful in any future global analyses of neutron scattering data to determine $b_{ne}$ and bound $\alpha$ and $\lambda$. We also discuss how various neutron interferometric and scattering techniques with cold and ultracold neutrons can be used to improve the precision of $b$ measurements and make some specific proposals., Comment: 13 pages, 1 figure