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New measurements of the rare α decays of natural Hf isotopes have been performed using four Cs$_2$HfCl$_6$ crystal scintillators operated in the low-background setup of DAMA/CRYS at the Gran Sasso National Laboratory (LNGS) of INFN, Italy. The “source = detector” approach provides high sensitivity to ultra-long-lived processes by embedding the decaying nuclei directly within the detector medium. After a detailed characterization of the detector performance, including assessment of the energy calibration and pulse shape discrimination (PSD) capability, data collected over 85.23 days have been analyzed. A precise determination of the half-life of the α decay of $^{174}$Hf to the ground state of $^{170}$Yb has been obtained: T$_{1/2}=(2.95\pm0.16)\times10^{16}$ yr, improving the accuracy of previous measurements and confirming their consistency. Particular attention, in this analysis, has been devoted to the evaluation of background contributions, especially from the possible $^{147}$Sm contamination, to a proper description of the shape of observed alpha peaks, and to careful assess the systematic contributions. No statistically significant signals have been observed for the $\alpha$ decays of other naturally occurring Hf isotopes ($^{176, 177, 178, 179, 180}$Hf), and new lower limits on their half-lives in the range 10$^{19}$-10$^{20}$ yr have been established. These results confirm the potential of Cs$_2$HfCl$_6$ crystal scintillators as a detector medium for rare-event searches and provide relevant a input for nuclear structure models of the extremely long-lived $\alpha$ emitters.