One of the more practically complex aspects to being both egalitarian and invested in head-covering is combining covering my head and putting on tefillin in a halakhically valid manner. If you're not willing to show some amount of hair, this just can't happen in public, because the head tefillin -both the bayit and the strap- need to rest on your head, not on another object (a mistake I see on Purim just about every year, from people with wigs/hats as parts of their costume, who just aren't thinking about it.)
I manage it by pushing my scarf back just a little bit, so that the tefillin rest in the appropriate place, with the bottom edge just at my hairline. Then I run my fingers around my scarf, pushing it back just enough so that the retzuah- the strap of the tefillin- is on my hair and not on my scarf. Since I use kippah clips to keep my scarves on, I lift them enough to let the retzuah slide underneath them. The whole thing adds about 5-10 seconds to how long it takes to put on tefillin, and doesn't muss most scarf arrangements too badly.
Some styles are better for davening in the morning than others- things like a bun style, braid, or pretty much anything that doesn't have much bulk on top will work with tefillin. When I want to do my tichel in a crown or other top-heavy fashion, I often just either wear a large, crocheted cap/kippah (if I'm davening at home, which has been my default this year- hopefully I'll make improvements to my minyan attendance soon), or tie my tichel just at the basic level for coverage and leave the tails hanging until after I finish davening.
On the other hand, I don't know how to work this sort of thing with a hat with any sort of brim, unless you can take it off to put the tefillin on and put it back on over them. So berets are doable with this system, but anything else gets tricky, unless you don't mind people getting a view of your hair, if you're in public.
Recap in step form:
1. Put on headcovering. Expect anything with a lot of bulk around your head to get jostled off or get in the way.
2. Put on tefillin in the appropriate fashion. When you get to the head, start putting it on, and say the bracha. Then:
3. adjust scarf out of the way of the tefillin in front.
4. Run your fingers around the strap, adjusting scarf out of the way. Settle everything in place.
5. Go on with putting your tefillin on, daven, etc.
1. When you come to remove your tefillah shel rosh (head tefillin), do so carefully, taking the strap out from under any clips or other obstructions first, for minimum frustration.
2. Touch up your tichel in any way you feel needed. For example: tug the front forward a little if you want, re-clip to tighten if needed, or put up the ends of your scarf, or additional scarves, to give yourself some height.