Just to build on your idea, at the end of interviews, I also recommend my clients actually TELL the interviewer you want to have the opportunity to make it to the next stage. It could sound something like this (after your last question):
“Thanks so much for giving me this opportunity to interview for this role. I just wanted to close by just reiterating how much I’d love to receive an offer to work for your company.”
The ending could be phrased as “opportunity to be invited back for the next round of interviews” or “chance to come in for the case interview round” or whatever the next stage may be.
Having interviewed many candidates myself in the past, I’m surprised how few candidates do this. It reiterates your enthusiasm and certainty about wanting to progress, and leaves a positive sentiment with the interviewer.