  Recently, methanol was identified as a sensitive target system to probe
variations of the proton-to-electron mass ratio $\mu$ [Jansen \emph{et al.}
Phys. Rev. Lett. \textbf{106}, 100801 (2011)]. The high sensitivity of methanol
originates from the interplay between overall rotation and hindered internal
rotation of the molecule -- i.e. transitions that convert internal rotation
energy into overall rotation energy, or vice versa, give rise to an enhancement
of the sensitivity coefficient, $K_{\mu}$. As internal rotation is a common
phenomenon in polyatomic molecules, it is likely that other molecules display
similar or even larger effects. In this paper we generalize the concepts that
form the foundation of the high sensitivity in methanol and use this to
construct an approximate model which allows to estimate the sensitivities of
transitions in internal rotor molecules with $C_{3v}$ symmetry, without
performing a full calculation of energy levels. We find that a reliable
estimate of transition sensitivities can be obtained from the three rotational
constants ($A$, $B$, and $C$) and three torsional constants ($F$, $V_3$ and
$\rho$). This model is verified by comparing obtained sensitivities for
methanol, acetaldehyde, acetamide, methyl formate and acetic acid with a full
analysis of the molecular Hamiltonian. From the molecules considered, methanol
appears to be the most suitable candidate for laboratory and cosmological tests
searching for a possible variation of $\mu$.
