 If you liked
Sora wa Akai Kawa no Hotori
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 ...then you might like
Legend of Nereid
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Innocent yet strong and capable girl gets thrown into one bad situation after another. Both protagonists fall in love with a man of nobility.
The difference is that Red River has no gender-bender and Legend of Nereid no time-travel. Furthermore, Legend of Nereid is a manhwa.
But mostly, if you enjoyed Red River (like I did) you should also enjoy Legend of Nereid.
Another story with a historical fantasy setting. Both have similar heroines and political/royal intrigue.