We present high-resolution measurements of the relative length change as a function of temperature of the organic charge-transfer salt $kappa$-(BEDT-TTF)$_2$Hg(SCN)$_2$Cl. We identify anomalous features at $T_g approx,63$ K which can be assigned to a kinetic glass-like ordering transition. By determining the activation energy $E_A$, this glass-like transition can be related to conformational degrees of freedom of the ethylene endgroups of the organic building block BEDT-TTF. As opposed to other $kappa$-(BEDT-TTF)$_2X$ salts, we identify a peculiar ethylene endgroup ordering in the present material in which only one of the two crystallographically inequivalent ethylene endgroups is subject to glass-like ordering. This experimental finding is fully consistent with our predictions from $ab,initio$ calculations from which we estimate the energy differences $Delta E$ and the activation energies $E_A$ between different conformations. The present results indicate that the specific interaction between the ethylene endgroups and the nearby anion layers leads to different energetics of the inequivalent ethylene endgroups, as evidenced by different ratios $E_A/Delta E$. We infer that the ratio $E_A/Delta E$ is a suitable parameter to identify the tendency of ethylene endgroups towards glass-like freezing.
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