Hello again Remi :).
First I have to say, to be honest, I would have taken the watch back to the shop to be repaired under guarantee if it was less than a year old. But credit to you for figuring out how to fix it.
I am not a watch expert but have managed to fix a much older quartz watch by trial and error (funnily enough a Tissot).
My only guess to try would be to see if the hour hand has come loose from its pinion. If the rest of the mechanism is working then maybe the movement is OK. If you wind the crown as if setting the time, all the way round 24 hours, does the date move? If so, maybe the movement is OK.
Good luck if you have to change the movement. If you do, you will have to remove the hands anyway so worth checking that the hour hand isn't loose.