HOW TO WEAR YOUR HOOD DOWN IN TOTK (SPOILERS)
So I don't know if you can actually get this another way, but I managed to find how you can wear the Hylian Hood down pretty much by accident and after a LONG process, so I decided to share it here.
To do this, you first have to go to Hateno Village and visit the new clothing shop. Here, you will find the famous mushroom seamstress Goutier (her french name. Idk if it's the same in english), which will launch a quest: basically, Mushroom Seamstress and the mayor of the village, Baldinn (again, french name) have beef and decide to organize elections to see who has the best vision to lead the village.
You will then need to do the three quests necessary to get to the election day, which, fair warning, will take a while. To start the two other quests, talk to the sister/assistant of Mushroom Seamstress and to the Mayor's wife (you can likely find her in Link's old house, cleaning)
Once it's all said and done and with the election over, you'll now get to visit the clothing shop again to receive your reward, which will also unlock the ability to buy other clothes there as well.
Buy the Hylian Hood if you don't already have one to wear it, and talk to Mushroom Seamstress. You will unlock a special dialogue where she says that she can't really see your face like this, and will offer to pull you hood down.
And we're there ! You can now wear your hood down, which is cool when it doesn't rain. To wear your hood up again, just buy another one and switch between the two items.