My husband and I visited Israel for our honeymoon a year-and-a-half ago. It was January,and therefore the middle of winter, so I brought hats, thinking two things: it may be cold, and I may want to cover my hair.(continued at link)
The connection between hats as a head-covering and hats as a hair-covering is not a strictly Jewish tradition. Covering the head in religious or holy spaces spans many religions. In Judaism, men are traditionally required to cover their heads. Women, in more Orthodox sectors also have a custom to cover their heads, but the custom relates more to covering the hair rather than the head. I know, a seemingly small distinction, but an extremely powerful one. Both are related to modesty in the presence of G-d, but men’s modesty and women’s modesty have distinctly different dimensions in Jewish life. Men are honoring G-d through wearing a kippa. Women, covering their hair and bodies, are honoring G-d.
photo here of a headcovering market stall from the wikipedia article on Meah Shearim, the ultra-orthodox neighborhood in Jerusalem the author mentions visiting.
If you're interested, please don't forget to click to tznius.com to learn more about tying scarves in one of a variety beautiful typically Jewish styles. I personally think these styles are lovely, and have used a couple my Christian self. I copied and pasted the two photos of women in headscarves from this site - full credit for these photos and styles goes to tznius.com.
Or maybe you'd like to make your own "snood" (or "caul") - which is like a headcovering with a pocket in the back to put your hair in, and is also often worn by Orthodox women (and old fashioned ladies) of various backgrounds. Check out "A TikkunKnitter’s Miscellany": "Continuing Cover: if I could snood".