  Probing the $\alpha$ cluster of $^{16}$O with the relativistic
$^{16}$O+$^{16}$O collisions has raised great interest in the heavy ion
community. However, the effects of the $\alpha$ cluster on the soft hadron
observables vary largely among different studies. In this paper, we explain the
differences by the compactness of the $\alpha$ cluster in oxygen, using
iEBE-VISHNU hydrodynamic simulations with different initial state $\alpha$
cluster configurations. We also find several observables, such as the intensive
skewness of the $[p_{\rm T}]$ correlator $\Gamma_{p_{\rm T}}$, the harmonic
flows $v_2\{2\}$, $v_2\{4\}$, $v_3\{2\}$, and the $v_n^2-\delta[p_{\rm T}]$
correlations $\rho(v_{2}^{2}, [p_{\rm T}])$, $\rho(v_{3}^{2}, [p_{\rm T}])$ in
$^{16}$O+$^{16}$O collisions are sensitive to the compactness of the $\alpha$
cluster in the colliding nuclei, which can be used to constrain the
configurations of $^{16}$O in the future. Our study serves as an important step
toward the quantitative exploration of the $\alpha$ cluster configuration in
the light nuclei with relativistic heavy ion collisions.
