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.