Hello! The saying "hat in hand" is usually used when we have to face up to something in a very humble way (to apologize or to ask for a favor). For example; if you are going to apologize to someone or ask for forgives for doing something wrong then you would say "I went to him hat in hand and apologized for being so mean." or "I went to them hat in hand to ask for a donation for the school." This saying comes from the old tradition of removing ones hat to show humble respect or honor for someone or something. When we use the word "proverbial" before a phrase (like hat in hand) it is clarifying the fact that we are referring to an idiom - in this case "hat in hand". In the text the writer says "proverbial hat in hand" because she is not actually standing with her hat in her hand but that she is referring to the saying "hat in hand" meaning she is humbly going to be asking for forgiveness for something or ask for a favor - it depends on the context of the story which I'm not too sure about,. I hope that helps!

I feel like there's some context here that I don't understand. "Hat in hand" usually means they want something, or they're begging for something. "He stood there, hat in hand, waiting for a response to his request for help". It means they are desperate and are begging for help. I guess it comes from the idea of a begger holding their hat out for people to put money in. When someone uses the word proverbial they are simply emphasising the use of the proverb, often without saying the entire proverb. For example, "it was the proverbial bird in the hand". This is from the proverb "a bird in the hand is worth two in the bush", meaning it is better to definitively have something than potentially have more.

"Hat in hand" - sometimes "cap in hand" - is a little, old-style idiom for "asking for a favour". So a proverb is a standard saying or phrase. She doesn't have a hat. She is wearing what sounds like a party dress. "Proverbial" here is used to mean this: she is not holding a hat, but she feels as if she is asking for his attention as some sort of favour, since she is all dressed up and he isn't. The balance of the potential romantic relationship is disturbed, and not in her favour. So she wishes she had changed into day clothes first, so that she could have maintained parity in the friendship. https://idioms.thefreedictionary.com/hat+in+hand