The median $ν_Γ(α)$ of the gamma distribution has no closed form, but its behavior at both ends is known: it is asymptotic to $2^{-1/α}e^{-γ}$ (with $γ$ being the Euler-Mascheroni constant) as the shape parameter $α\to 0$, and approaches $α- 1/3$ as $α\to \infty$. We write approximations as interpolations $(1 - g)ν_0 + gν_\infty$ between two elementary-function bounds or approximations, each exact in one limit, with an interpolator $g(α)$ going from 0 to 1, and compare three such forms: a first one between previously proven bounds, a second ending in an approximation from the beta median that is exact at $α= 1$ but does not quite approach $α- 1/3$ as $α\to \infty$, and a third one, between bounds proved here, that corrects the large-$α$ end of the second. For each form, the ideal interpolator's slope at 0, approach to 1, and value at $α= 1$ are derived in closed form. Rational interpolators with every coefficient set in closed form from these properties, with no numerical fitting, reach a worst relative error of 0.0011 over all $α> 0$, with absolute error below 0.0002. Fitting some of these or other coefficients numerically to minimize the worst absolute-or-relative error then trades accuracy against complexity: the Pareto frontier reaches 0.000087 with two fitted constants and 0.000041 with five, and from two on uses degree-4 rationals of the second form with no $α^2$ term in the numerator. The proofs that the end functions of the first and third forms are bounds are machine-checked in Lean 4.