Precise atmospheric observations have been made for a growing sample of warm Neptunes. Here, we investigate the correlations between these observations and a large number of system parameters to show that, at 95% confidence, the amplitude of a warm Neptune's spectral features in transmission correlates with either its equilibrium temperature (Teq) or its bulk H/He mass fraction (fHHe)-in addition to the standard scaling. These correlations could indicate either more optically thick, photochemically produced hazes at lower Teq and/or higher-metallicity atmospheres for planets with smaller radii and lower fHHe. We derive an analytic relation to estimate the observing time needed with JWST/NIRISS to confidently distinguish a nominal gas giant's transmission spectrum from a flat line. Using this tool, we show that these possible atmospheric trends could reduce the number of expected TESS planets accessible to JWST spectroscopy by up to a factor of eight. Additional observations of a larger sample of planets are required to confirm these trends in atmospheric properties as a function of planet or system quantities. If these trends can be confidently identified, the community will be well-positioned to prioritize new targets for atmospheric study and eventually break the complex degeneracies between atmospheric chemistry, composition, and cloud properties.