R. Moshe Feinstein (March 3, 1895 – March 23, 1986) comments on the halakhic permissibility of toupees, and on their interactions with tefillin.
מי שקרח ראשו אם מוטר להניח שערות זרות ודינם בענין חציצה בתפילין
אני לא שמעתי ששייך דבר כזה שלמי שקרחו שערותיו יהיה שייך שיצמחו שם שערות זרות שיניחו על הראש, ואם איכא דבר כזה היה זה דבר מותר לעשות, ולענין חציצה לתפילין קודם שנצמחו אין זה חציצה כיון שאי אפשר לחסירם וכל מי שאין לו שערות לא יסירם, ואם ליכא דבר כזה אבל הוא רק דביקת השערות זרות להגוף ע”י דבק נמי כיון שכן ישאר לעולם ורוצה בזה לא יחצוץ לתפילין, ואם הוא פאה נכרית שיכול להסיר בכל עת שרוצה אף שהוא לנאותו שהוא בוש מלגלות ראשו בבהכ”נ יניח את של ראש בלא ברכה וכשיבא לביתו יגלה ראשו ויניח תפילין בברכה
Someone who is bald: Whether it is Permitted to Wear False Hair, and the Law in the Matter of חציצה and Tefillin.I have not heard that a thing like this is possible for someone who has lost their hair- that foreign hair that they place on their head will then grow there, but if there is such a thing, it is permissible to do. As for the matter of separation (חציצה) for tefillin before they [the hairs] are grown, this is not a separation since it is impossible to remove them, and anyone who does not have hair does not remove them. And if there is no such thing, but rather he just attaches the foreign hair by means of glue, similarly, since they remain forever, and he wants them- they are not a separation (חציצה) for tefillin. And if it is a wig that he can remove at any time that he wants, even if it is for aesthetic purposes, and he is embarrassed to reveal his head in the synagogue, he may put on the head-tefillin without a blessing, and when he gets home, he reveals his head and puts on the tefillin with a bracha.
The first thing that I noticed, in this teshuvah, is that one may delay putting on tefillin in the proper way in order to pray with a minyan without embarrassment (בושה). I would then suggest that this is even more so the case when the issue one is dealing with is tzniut/communal standards rather than individual embarrassment without many religious implications. [A little research into textual views of baldness reveals Mishnah Bechorot 7:2, which disqualifies a totally bald kohen (without even a fringe around the edge of the head) from service in the Temple and Bava Kama 60b, where there is the story of a man with two wives- one plucks out his dark hair and the other plucks out his white hair, leaving him bald. However, these don't give a strong implication of shame or inappropriateness about the baldness itself, although the first does cast it as a מום, an imperfection of the body.]
So a woman who does not want to reveal enough of her hair to put on tefillin with all the relevant pieces of the tefillin touching her hair/head directly could put them on over her covering, as long as she put them on without it either before or afterward, in private. Even when one is comfortable showing that much hair (not actually all that much hair if one is wearing a scarf, once you've practiced a little bit- but plausibly a little more than a tefach according to R. Soleveitchik (the two-finger measure, rather than the four-finger measure... I should really find out where he writes that and share it here, at some point.)), getting the tefillin around one's covering can be complicated, depending on style. The chance to just not worry about it and put on tefillin without the whole arrangement (in my case, with a kippah or cap instead of a scarf) gives an additional option for managing one's day, and for praying in public on a weekday.
It also presents an interesting and surprising interaction between one's feelings and one's halakhic obligations. Now I don't Like davening in the morning without my tefillin, but have definitely put them on before davening, then taken them off and gone to shul, when there's a reason for me to go to a shul/minyan where it would be uncomfortable for me and for the people around me for me to wear them. This is a pretty similar circumstance, and R. Moshe seems quite comfortable with the situation. It's a model of handling conflicting needs- everything gets fulfilled, just with some delay. It is significant because it treats emotional needs as halakhically significant, in justifying non-ideal mitzvah fulfillment.
It's more remarkable because it's letting someone look like they're fulfilling a mitzvah when they are not, and plausibly runs into an issue of marit ayin. That issue would be more significant for someone wearing a scarf, since a toupee might not be known to be a toupee, while a scarf is obviously not a part of the person. That does detract from my comfort in apply R. Moshe's teshuvah for wearing a tichel with tefillin. It would work better for someone wearing a wig, (who would likely need it more- I don't know how one would manage a wig and tefillin at the same time in a kosher manner).