It's an interesting question and as you suggest in your title and first sentence, the subject is only hinted at.
The first part of your question is, "Where in the Tanakh is there support for females (no matter how old) ever having their hair or head uncovered?"
This is actually hinted at in two places but requires a small introductory explanation.
Within the allegories used in the Torah, the Female is compared to the Earth (as in soil, like the expression Earth-Mother) like is explained in Sefer Me'orot Natan, Letter Aleph, Chapter 126, ארץ. This relates to earth/soil being the material source of the power of endless fructification. That the Feminine is in character, Fructuous, full of and producing fruit, so to speak.
In parshat Bereshit, there are paradigms of Creation established which repeat like fractals across the six days of Creation and throughout the entire universe even to the present. So the first paradigm introducing the distinction between Masculine and Feminine is the appearance of the dry land from the waters in Bereshit 1:9.
The waters are initially one and then are separated into two by the Rakia. And then from the Lower Waters comes forth the dry land (יבשה) which will be called Earth. And the Waters will be called Seas.
The Earth corresponds with the Feminine aspect and the Seas correspond with the Masculine. In this case, the Seas are an allegory to the Semen which flows ultimately to water the Garden which is comprised of the dry land.
And this same paradigm repeats within Creation in Bereshit 1:27, 2:7-15 and 2:21-24. This process of repeating is called Seder Hishtalshelut (סדר השתלשלות) in Kabbalistic literature.
With that introduction, it is now possible to point to the hint of uncovered hair with females. That growing and showing of their hair is equated with the appearance of the Desheh, (the concept of Malchut like is discussed in Sefer Kehillat Yaacov, Erech דשא) over the Earth in Bereshit 1:11-12. That this is their covering and beauty and an outward sign of their being matured, prepared and ready for a mate and reproduction.
And this same language pertaining to the Desheh is found in Tehillim 23:2. This is the same chapter of Tehillim that many say at each of the three meals of Shabbat in hinting to the elements which make-up a Jewish marriage. That there are three partners in a Jewish marriage.
The first 45 words of the chapter, lines 1 through 5, hint to Adam (אדם) which is gematria 45. The next 3 words in posuk 6 (אך טוב וחסד) have the ending letters כב״ד, (which also refers to the liver כבד, the organ that causes the blood to thicken and become more material similar to the womb) which is the gematria of G-d's name (יהוה). This refers to the middle partner, G-d, in a marriage like is referenced in the Talmud. And the final 9 words allude to the Feminine, the wife, who corresponds with the letter Tet (ט) whose form is compared to an empty vessel prepared to receive, like is stated in Sefer HaBahir, chapter 124.
This chapter of Tehillim is also recounting the episode from Bereshit when Chava, Adam's mate was walking in the Garden just prior to her encounter with the Nachash in Bereshit 3:1-6. And at that time, she was rejoicing in the beauty of her hair (בנאות דשא ירביצני) and contemplating her future Motherhood with her betrothed mate.
This seems to be implied from the understanding brought in connection with Ketubot 72a which connects the mitzvah of the suspected Adulteress and the ritual involving uncovering her hair to actually untying her braids like is brought by Rashi on BeMidbar 5:18. Braided hair is a minhag of modesty for unmarried Jewish women like is discussed in Magen Avraham 75:3, Mishnah Berurah 75:12, 303:82 and at length in Sefer Oz v'Hadar Levushah (Modesty: An Adornment for Life by Rabbi Pesach Eliyahu Falk, A"H) Sec. 5H, pp. 262-265. And so this detail of the chapter in Tehillim would actually be indicating that Chava was betrothed to be married but had not yet had relations, conceived and had children. And this also follows the explanation of Chacham Yosef Chaim, the Ben Ish Chai in his commentary to Shas, Ben Yehoyada to Yoma 47a:4 who stays:
ובזה מובן הטעם מה שברא הקב"ה את חוה בקליעת שער ולא בראה שערה סתור
The second part of your question is more difficult. You asked, "Where in the Tanakh is it extrapolated that females are to cover their heads or hair when praying or reciting the name of HaShem?"
This is derived from Bemidbar 5:11-31 discussing the subject of the Suspected Adulteress. In BeMidbar 5:18 it emphasizes that the hair of the wife suspected of adultery will be uncovered as part of the ritual to be performed.
What is emphasized here is that the normal state of being for a married Jewish woman is for her hair to be covered at all times. And when her hair is uncovered, she is considered to be naked. But prior to a wife having her first physical relations with her husband, merely having her hair in braids is sufficient like was the case with Chava, Adam's wife.
And it is in the general context that any Jewish individual, meaning male or female, is forbidden to utter Holy words while naked, that the prohibition would be learned out for females too. And this prohibition is learned from Devarim 23:15 which states:
כִּי֩ יְהוָ֨ה אֱלֹהֶ֜יךָ מִתְהַלֵּ֣ךְ ׀ בְּקֶ֣רֶב מַחֲנֶ֗ךָ לְהַצִּֽילְךָ֙ וְלָתֵ֤ת אֹיְבֶ֙יךָ֙ לְפָנֶ֔יךָ וְהָיָ֥ה מַחֲנֶ֖יךָ קָד֑וֹשׁ וְלֹֽא־יִרְאֶ֤ה בְךָ֙ עֶרְוַ֣ת דָּבָ֔ר וְשָׁ֖ב מֵאַחֲרֶֽיךָ׃
That we are prohibited from having any kind of nakedness found in regard to us when G-d's presence is revealed in our domain and proximity.
