Scout absolutely hates the various attempts made by her friends and relatives to make a lady out of her. Her favorite clothing is her precious overalls, and she resents having to wear anything else. When Calpurnia takes Jem and Scout to her church, she makes sure that Scout is properly dressed, including a petticoat and a pink sash. She complains that the dress "came up like a tent" when she sat down. When the ladies of the Missionary Circle joke about the whereabouts of Scout's "britches," she tells them honestly that they are "under my dress." When Jem tells her that it's time for her to "start bein' a girl and acting right," Scout "burst into tears."
To me, the clearest example of this comes early on in the book, in Chapters 4 and 6. I suppose you could say these are examples that show that she wants to be included in what the boys do, rather than being excluded because she's a girl. I think that counts as being self conscious.
What Scout says that shows me this is in Chapter 6. There, Jem and Dill want to go peek at Boo Radley but Scout does not. But when Jem tells her she is acting like a girl, she says
With that, I had no option but to join them.
She had already reported, back in Chapter 4, that Jem was saying stuff to her about her acting too much like a girl. Here, she responds in a way that makes me think she's self-conscious about it.
During the mid 20s chapters, at the Missionary Society Tea Party, women tease Scout about what she might be wearing. They notice her in her dress, and wonder if she has her pants on underneath because that is all that they ever see her in.
Miss Maudie comes through for Scout in this moment. She gives Scout's hand a squeeze at the tea letting Scout know she is there for her. Scout had no idea how to react but tells the truth nonetheless and answers that she doesn't have any pants on underneath at all.