Finally, I've met Mark/PT's tefillin challenge
Okay, here's the story:
1. My understanding is that halachah (Jewish religious law) requires that the tied-together strap of the shel rosh/head tefillin (singular: tefillah?) be worn directly on the head. Therefore, it would not be permissible to wear the head tefillin over a shaitel/sheitel/wig (or one of those partial wigs called a fall).
2. At the other extreme, a kippah/yarmulke/skullcap, though frequently worn by non-Orthodox women and easy to wear with head tefillin, would probably be considered a man's garment (beged ish), and, therefore, a no-no, by most Orthodox women, since halachah forbids either gender from wearing a garment of the other gender.
3. Trust me on this one: It's not physically possible to shift a hat around on, or just above, one's head (to keep one's head covered) with one hand while putting on the head tefillin with the other hand. I know, because I've tried it many times, without much success. One really does need two hands to put on head tefillin. I'll have to save my new "lampshade" hat for Shabbat (Sabbath) and Yom Tov (Festival), when one does not wear tefillin.
4. While it's actually fairly easy to put the head tefillin on over a snood--all one has to do is to readjust the snood afterward to ensure that no part of the snood is underneath the tied-together tefillin strap, which must lie directly against the head--I'm not bonkers about wearing a snood in public because, personally, I find most snoods rather unattractive. See these examples of snoods, and judge for yourself.
The solution, and my response to Mark's challenge, is to wear a pre-tied tichel/mitpachat/scarf. The elastic on a pre-tied scarf keeps the scarf from falling off while one is putting the tefillin on over one's head, thus enabling one to use two hands to put on the shel rosh while still keeping part of one's head covered. As with a snood, one must readjust the scarf to ensure that the tefillin strap lies directly against the head, with none of the fabric caught underneath the strap.
Don't bother trying to tie the loose ends, if any, of your pre-tied scarf in any fancy manner, whether by bringing them around to the top of the head in front and tying a loose knot (or winding them together, turban-style) or by tying them in a bun in back, as is the current fashion. They'll get in the way and/or unravel while you're shifting the scarf around to get it out from under the tied-together tefillin strap. Just take the ends and tuck them under the elastic in the back of the scarf, to get them out of the way.
Once you've gotten the scarf's loose ends tucked away, make the brachah (blessing) over the head tefillin, place the bayit (box containing the biblical quote hand-written on parchment) at the front of the head, use both hands to put the tied-together part of the strap around the head--that's the part that's impossible to do one-handed, while wearing, and trying not to remove, a hat--and pull one untied part around each side of the neck and to the front. Then, holding the front of the scarf so that it won't fall off, move the tied-together part of the strap and the sides and back of the scarf as necessary to ensure that none of the scarf is caught under the strap, which must rest directly against the head. Finally, make sure that the bayit is in its proper place, centered, with the front lower edge above the hairline, and that the knot at the back of the head tefillin is centered at the back of the neck, just below the skull. (To the best of my knowledge, this is the halachically-required positioning of the head tefillin. Please do correct me if I'm wrong!) I believe that it's permissible to pull the scarf back over the straps, if you'd like to keep more of your head covered. (Again, corrections requested, if necessary.)
Here's the way it looks on me. (Nice photo of the shel rosh and scarf. Not such a good photo of the woman wearing them. Oh, well.)
An Orthodox married woman who believes that a married woman must cover some, but not necessarily all, of her hair in public may find this an acceptable solution for putting on tefillin in public while maintaining her tzniut (modesty), assuming that she is interested in putting on tefillin. As far as I can determine, an Orthodox married woman who believes that all of her hair must be covered in public would not be able to lay tefillin in public at all, but could do so in private, if she chose. (In addition, there's the upper-arm issue: Since nothing is permitting to come between the shel yad/hand tefillin and the arm and hand themselves, a woman whose interpretation of the laws of tzniut includes a prohibition against revealing her arm above the elbow would have to lay tefillin in a private place, since the bayit (box containing the biblical quote hand-written on parchment) of the shel yad/hand tefillin must be placed above the elbow.)
There are, of course, many women in the Jewish community who are of the opinion that a woman's traditional role in Judaism is quite honorable as it is, and feel no need for externals. That role has been the way of our mothers for generations, and has my respect, though it is not a role that I, myself, am comfortable following.
On a related note, you may be surprised to know that Jen Taylor, a soferet (scribe) and designer of Tefillin Barbie, does not think that all Jewish women should wear tefillin--and you may be even more surprised to know that I agree with her. I think she makes a compelling case.
Look, Shira, you're never going to get the tfillin to fit over your big-hair shaitel anyway, so let's just drop the whole thing.
Ouch! Truth to tell, I never even thought about how traditional married women who cover (at least part of) their hair in public were supposed to fit tefillin over a wig, hat, or scarf. Score another point for the tzniut [modesty] patrol. (: