Gravity-driven fingers in uniform porous medium are known to have a distinctive nonmonotonic saturation profile, with saturated (or nearly so) tips and much less saturation in the tails. In this work, constant-flux infiltrations into confined porous media (laterally smaller than the finger diameter, thus essentially one-dimensional) are found to produce saturation overshoot identical to that found in gravity-driven fingers. Light transmission is used to measure the saturation profiles as a function of infiltrating flux, porous media grain size, grain sphericity, and initial water saturation. Saturation overshoot is found to cease below a certain minimum infiltrating flux. This minimum flux depends greatly on the grain sphericity and initial water content of the media and slightly on the mean grain size. The observed saturation overshoot is inconsistent with a continuum description of porous media but qualitatively matches well observations and predictions from discrete pore-filling mechanisms. This suggests that pore-scale physics controls saturation overshoot and in turn gravity-driven fingering.