  Experiments aimed at searching for variations in the fine-structure constant
$\alpha$ are based on spectroscopy of transitions in microscopic bound systems,
such as atoms and ions, or resonances in optical cavities. The sensitivities of
these systems to variations in $\alpha$ are typically on the order of unity and
are fixed for a given system. For heavy atoms, highly charged ions and nuclear
transitions, the sensitivity can be increased by benefiting from the
relativistic effects and favorable arrangement of quantum states. This article
proposes a new method for controlling the sensitivity factor of macroscopic
physical systems. Specific concepts of optical cavities with tunable
sensitivity to $\alpha$ are described. These systems show qualitatively
different properties from those of previous studies of the sensitivity of
macroscopic systems to variations in $\alpha$, in which the sensitivity was
found to be fixed and fundamentally limited to an order of unity. Although
possible experimental constraints attainable with the specific optical cavity
arrangements proposed in this article do not yet exceed the present best
constraints on $\alpha$ variations, this work paves the way for developing new
approaches to searching for variations in the fundamental constants of physics.
