So there was a manhua/manhwa I recalled reading about a month or two ago that I forgot to save to a reading list. Setting has a European ambience, but at any rate, the MC transmigrates into a girl who is mistaken for a battle slave and taken to a coliseum. She breaks free of her bonds during battle and escapes, crash landing into a high ranking noble's carriage. This noble decides to take her in when the coliseum owners demand she be returned. The transmigrated MC's most distinctive feature prior to joining the noble was a red jacket, I think Pretty vague, but that's all I can remember.
Found it after digging through my internet history: When I Traveled to Another World I Was Forced to Be the Princess of the City?