Devotional Head Covering Source: Kiddushin 8a

Translation courtesy of Sarah Mulhern, rabbinical student and friend.

Text:

 וכגון דקביל כהן עילויה; כי הא דרב כהנא שקיל סודרא מבי פדיון הבן, אמר ליה: לדידי חזי לי חמש סלעים. אמר רב אשי: לא אמרן אלא כגון רב כהנא, דגברא רבה הוא ומבעי ליה סודרא ארישיה, אבל כולי עלמא לא 

Translation:  

[What is the law in] a case where the priest accepts [an object of insufficient value in return for the redemption of a son] as happened in the case of R. Kahana, who accepted a scarf for a son's redemption, and said to him [the father],‘To me it is worth five selas.’? R. Ashi said: "This is only applicable in the case of a person like R. Kahana, who is a great man and needs a scarf /turban
for his head, but not of people in general."

Comments:

This text refers to head covering only in passing, as part of a social reality.  The focal point of the text is the question of how to handle an oddly done redemption of a firstborn son, which requires the use of a sum of 5 selas- usually this would involve 5 silver coins, contemporarily.  This text raises the question of relative worth- if something is worth more to you (the receiver) than it is normally, could it count?  The answer seems to be "Yes, but no"- in theory, it could, but we don't really want to go there.

However, the example is what makes this text relevant to our question.  The example is used to indicate that the specifics here make the results of this case is not generally transferable- because Rav Kahana is a particularly important person, therefore he needs a turban.  This suggests that turbans at least are a marker of social status, rather than of religiosity or piety.  (I notice that in stories that talk about head covering as piety in the Talmud, the covering is either of non-specific style or is a robe worn over the head, rather than being a turban.)

Turbans (much) later become the usual covering for men in most of the Middle East, but I'm not sure when that happens.