We report a measurement of the ensemble-averaged transverse spin relaxation time $({T}_{2}^{\ensuremath{\ast}})$ in bulk and few molecules of the organic semiconductor tris-(8-hydroxyquinolinolato aluminum) or ${\text{Alq}}_{3}$. This system exhibits two characteristic ${T}_{2}^{\ensuremath{\ast}}$ times: the longer of which is temperature independent and the shorter is temperature dependent, indicating that the latter is most likely limited by spin-phonon interaction. Based on the measured data, we infer that the single-particle ${T}_{2}$ time is probably long enough to meet Knill's criterion for fault-tolerant quantum computing even at room temperature. ${\text{Alq}}_{3}$ is also an optically active organic, and we propose a simple optical scheme for spin qubit readout. Moreover, we found that the temperature-dependent ${T}_{2}^{\ensuremath{\ast}}$ time is considerably shorter in bulk ${\text{Alq}}_{3}$ powder than in few molecules confined in 1--2-nm-sized cavities. Because carriers in organic molecules are localized over individual molecules or atoms but the phonons are delocalized, we believe that this feature is caused by phonon bottleneck effect.