Meaning of "stared at its lining, touched the inner lining"?

Hello everyone. I am reading a novel, and I came across this expression. Could you please let me know its meaning?

Clara had left her coat on her seat. I let my hand rest on her coat, stared at its lining, touched the inner lining. Clara. It was also my way of remembering I was not alone, that she would very shortly come back and take up her seat again and tell me—or perhaps not—why she had taken so long. Sometimes just placing my coat next to my seat when I am alone in a movie theater is itself a way of conjuring a presence in the dark, of imagining that someone has stepped out for a second and will any moment come back—which is what happens in the dead of night, when those who have left our lives suddenly lie next to us no sooner than we’ve whispered their name into our pillow. Clara, I thought, and there she’d be, taking the seat next to mine.

- André Aciman, Eight White Nights, Second Night

This is a novel published in the United States of America in 2010. This novel is narrated by the nameless male protagonist who meets Clara at a Christmas party in Manhattan. The next day, the protagonist runs into Clara standing in front of the movie theatre. (He had mentioned to Clara during the party of the previous night that he would go to the Rohmer festival.) So they go to the movies together, and during the intermission when Clara went outside for a phone call, the protagonist touches the coat she left at her seat.

Here, this is just my small question, but I wonder what the underlined expression means.

I think the protagonist first saw the inner cloth of the coat and touched it... But first it is "lining" and then it is "inner lining," so I wonder how "lining" and "inner lining" are different.

My guess is that "inner lining" might be an innermost part of the lining (=the cloth sewn inside the outer coat cloth, which comes in contact with the skin), but I am not sure...

Thank you very much for your help.