The word can mean a covering in general, or a head-covering in particular. In the Concordance, I only found three places in תנ"ך where the word appears in this form-
For example, ישעיהו ט:יג (and repeated later in ישעיהו):
וְזָנָב כִּפָּה וְאַגְמוֹן יוֹם אֶחָד
The Chabad translation, following Rashi translates it as "arch", and Artscroll translates it as "canopy".
Rashi writes:
לשון מלכים ושלטונים כפה ארקוול"ר בלעז כלומר אותם החופפים עליהם ככיפה
זו
An expression denoting kings and sultans. כִּפָּה, arcum voltum,
arvolt in O.F. [an arch] i.e., those who hover over them like an
arch. (Chabad translation)
On the other hand, the מצודות (and the מלבי"ם) relate the word to branch:
מצודת דוד- השרים הדומים לענף יפה
מצודת ציון- כן נקרא הענף כמו וכפתו
לא רעננה (איוב טו:לב) לפי שהיא כפופה
This posuk in איוב is the other place where the word כיפה appears. Here's what Rashi writes there:
כפיית ענפיו כל ענפי אילן דומין לכיפה
And here's what the מצודת ציון writes:
ענפיו כמו כפות תמרים (שם כג) ע"ש שהמה כפופים
So we have here a close relationship between "canopy/arch", "branch" and "bent".
Note that the ערוך has an two entries for כפה, one meaning anything circular and the other referring to a covering. Of course, the ערוך is referring to the language of the גמרא, but I believe this reflects the relationship I mentioned above.
Rashi uses the term arch, which is a semi-circular structure.This appears to follow from כפוף meaning bent- as opposed to straight. כף (as in כפות תמרים) means branch, which have a bent shape. (Indeed, "arch" derives from the Latin "arcus" meaning bow, which is both "bent" and a "branch"!) Also consider כף= spoon, which also has the same circular shape- see the ערוך. The letter כ itself is a semi-circle bent shape.
In summary, kippa- a head-covering derives from the word in תנ"ך that means canopy/arch/covering. That word is derived from branch/bent. While "palm" may be the ultimate etymology, the word כפה is not directly related to it, rather it follows from "bent" or "branch".