  By recasting the Klein--Gordon equation as an eigen-equation in the coupling
parameter $v > 0,$ the basic Klein--Gordon comparison theorem may be written
$f_1\leq f_2\implies G_1(E)\leq G_2(E)$, where $f_1$ and $f_2$, are the
monotone non-decreasing shapes of two central potentials $V_1(r) = v_1\,f_1(r)$
and $V_2(r) = v_2\, f_2(r)$ on $[0,\infty)$. Meanwhile $v_1 = G_1(E)$ and $v_2
= G_2(E)$ are the corresponding coupling parameters that are functions of the
energy $E\in(-m,\,m)$. We weaken the sufficient condition for the ground-state
spectral ordering by proving (for example in $d=1$ dimension) that if
$\int_0^x\big[f_2(t) - f_1(t)\big]\varphi_i(t)dt\geq 0$, the couplings remain
ordered $v_1 \leq v_2$ where $i = 1\, {\rm or}\, 2, $ and $\{\varphi_1,
\varphi_2\}$ are the ground-states corresponding respectively to the couplings
$\{v_1,\, v_2\}$ for a given $E \in (-m,\, m).$. This result is extended to
spherically symmetric radial potentials in $ d > 1 $ dimensions.
