  Recently it has been pointed out that diagonal Pad\'e approximants to
truncated perturbative series in gauge theories have the remarkable property of
being independent of the choice of the renormalization scale as long as the
gauge coupling parameter ${\alpha}(p^2)$ is taken to evolve according to the
one-loop renormalization group equation -- i.e., in the large-${\beta}_0$
approximation. In this letter we propose and describe an improvement to the
method of diagonal Pad\'e approximants. The improved method results in
approximants which are independent of the chosen renormalization scale when
${\alpha}(p^2)$ evolves at any chosen (in principle arbitrary) loop-level.
