This doesn't solve the underlying issue, but it's a temporary stop-gap to those who want their hoods down but don't want to wear a hair-hiding hat. There are a few orange moddable circlets and helmets in-game that will hide hoods while leaving your hair and/or most of your face showing.
Republic side:
- Heavy: Soverign Mind Headgear (Nar Shadda equipment commendations, level 25)
- Medium: Fervent Battle Headgear (Nar Shadda equipment commendations, level 25)
- Light: Consular's Headgear (Crafted by synthweaver, level 19. Not wearable by companions)
There are also several other synthweaver crafted ones, but these are the earliest you can get as far as I can tell.
Empire Side:
- Heavy: Reinforced Battle Headgear (Hides hair)
- Heavy: Tempered Laminoid Headgear (hides hair)
- Medium: RD-03B Coordination Headgear (Nar Shadda Equipment commendations, level 25, hides eyes)
- Medium: TD-03B Infiltrator Headgear (Nar Shadda Equipment commendations, level 25, hides eyes)
- Medium: Sniper's headgear (Armormech, level 19, hides hair)
- Medium: RD-07A Vendetta Headgear (Level 47, semi-rare random drop)
- Light: Channeler's Headgear (Crafted by synthweaver, level 27, hides hair)
- Light: Traditional Nylite Headgear (Crafted by synthweaver, level 23)
Note: there are many hats with this same make but this is the earliest one you can get.
Adaptive:
- Death's Claw helm (It covers a lot of the face unfortunately, but does show hair)
Unfortunately, a lot of the Empire side ones are kind of ugly. This is mitigated somewhat by the fact that a lot of the Imperial armor doesn't have hoods. However, Inquisitors get a lot of hoods, and not much option to hide them. They have lots of circlets but none of them hide hoods. The only decent one I've found for inquisitor is the Traditional Nylite headgear (and its clones) and it looks kind of strange with several of the hairstyles.
Would it be possible, if togglable hoods are difficult, to get a simple social hat that hides hoods for everyone? Maybe one that's completely invisible, or is a tiny circlet? Failing that, would it be possible to make some of the social armor hide hoods? Especially the House Ulgo circlet (social III) or the Voss Mystic Circlet (social V) since they're just circlets, or the Belsavis one since it's a slave collar.