Observation of precession of the equinoxes can most easily be seen over generational timespans by observation of the position of the full moon against the background stars against the markers of the seasons.
Precession, occurring at the very slow rate of one degree of arc per human lifetime (71.6 years) is often considered almost indetectable. The position of the sun cannot be seen against the background stars. However, for both ancient and modern observers, the ecliptic position of the full moon is easily seen, providing the most obvious marker of precession.
For example, in ancient Israel the Passover festival was celebrated in spring, at the 14th day of the month of Nisan on the night of the Paschal Full Moon after the northern vernal equinox. Before the time of Christ, the full moon on this date could be seen as in the constellation of Libra, meaning the sun was directly opposite, in the constellation of Aries. However, as the equinoxes precessed, the stars at Passover shifted, so that after the time of Christ the festival occurred when the moon was in the constellation of Virgo, and so the sun was in the opposite sign of Pisces. Passover dating is the same as for Easter. Now the full moon at Easter is nearly at the beginning of the constellation of Virgo, and will precess into Leo over the next centuries, as the Easter sun precesses from Pisces into Aquarius.
The ready observation of the shift of the Passover moon, and the widespread ancient knowledge of observational astronomy, suggests that knowledge of precession would likely have been more widespread in the ancient world than is textually attested due to its effect on the observed position of the moon. Through the 2000 years before Christ when the sun after the equinox moved through the constellation of Aries, the full moon for that month was in Libra. Passover was calculated by the ripening of barley, with an intercalary month added if the barley was not ripe. The simple observation of the stellar position of the full moon - in Libra - would have previously provided easy astronomical confirmation of the choice of month. This was no longer the case after Christ, because the Easter full moon now occurred in the constellation of Virgo. It seems plausible that ancient Jewish astronomers responsible for setting the date of Passover would have noted that a traditional observation of the paschal moon in Libra was moving out of alignment.
This popular cultural application of precession as visible in the position of the full moon against the seasons suggests that the traditional attribution of the discovery of precession to Hipparchus probably fails to recognise the likelihood of broader knowledge of this phenomenon. I am not aware of any documentation of this use of the moon, but it is an area of culture that is not well documented.
As a deductive conclusion, to what extent does awareness of the visible heavens in the ancient world suggests this observation of precession of the moon is likely?