Observations with the IRS spectrograph onboard Spitzer have found many sources with very deep Si features at 9.7 μm, that have optical depths of τ > 1. Since it is believed that a few of these systems in the local Universe are associated with Compton-thick active galactic nuclei (hereafter AGN), we set out to investigate whether the presence of a strong Si absorption feature is a good indicator of a heavily obscured AGN. We compile X-ray spectroscopic observations available in the literature on the optically-thick (τ9.7 μm > 1) sources from the 12 μm IRAS Seyfert sample. We find that the majority of the high-τ optically confirmed Seyferts (six out of nine) in the 12 μm sample are probably Compton-thick. Thus, we provide direct evidence of a connection between mid-IR optically-thick galaxies and Compton-thick AGN, with the success rate being close to 70% in the local Universe. This is at least comparable to, if not better than, other rates obtained with photometric information in the mid to far-IR, or even mid-IR to X-rays. However, this technique cannot provide complete Compton-thick AGN samples, i.e., there are many Compton-thick AGN that do not display significant Si absorption, with the most notable example being NGC 1068. After assessing the validity of the high 9.7 μm optical-depth technique in the local Universe, we attempt to construct a sample of candidate Compton-thick AGN at higher redshifts. We compile a sample of seven high-τ Spitzer sources in the Great Observatories Origins Deep Survey (GOODS) and five in the Spitzer First-Look Survey. All these have been selected to have no PAH features (EW6.2 μm < 0.3 μm) to maximise the probability that they are bona-fide AGN. Six out of the seven GOODS sources have been detected in X-rays, while for the five FLS sources only X-ray flux upper limits are available. The high X-ray luminosities (LX > 1042 erg s−1) of the detected GOODS sources corroborates that these are AGN. For FLS, ancillary optical spectroscopy reveals hidden nuclei in two more sources. SED fitting can support the presence of an AGN in the vast majority of sources. Owing to the limited photon statistics, we cannot derive useful constraints from X-ray spectroscopy on whether these sources are Compton-thick. However, the low LX/L6 μm luminosity ratios, suggest that at least four out of the six detected sources in GOODS may be associated with Compton-thick AGN.