Panicked a bit and gave a salary range in my first interview, wishing for a higher number now in interview #3. Can I still go for more?

Long story short, I applied for a long shot, dream job. I didn't even anticipate a response, but to my surprise, they called for interview #1 with HR (a less than 10 minute call). The salary range listed on the ad was $30k between the highest and lowest amounts. She asked me what I was thinking I'd make, and I threw out a "mid-x's" in the middle of the range - I know I should have been more vague, but she got me off guard.

Now that I've had a longer interview with the COO and am doing my final interview/very likely getting my offer (per the COO) this week, I'm wishing I had gone a little higher (about $5k more than what I said). How can I swing asking for a higher amount than my initial response without risking blowing the whole thing?

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