Applied at prestigious university for administrative position

I recently had a first-round interview for an administrative position at a prestigious university. During the interview, the recruiter asked me about my minimum salary expectations. I responded by referencing the pay range listed for the position, which was between $19 and $29 per hour, suggesting I was thinking around the midpoint. To my surprise, the recruiter then suggested a specific figure of $25, which caught me off guard. I ended up agreeing to that figure, but I didn't feel comfortable with it, especially since my current salary is $50k. The interview wasn't the most awkward I've experienced, but it definitely wasn't what I expected. To make things more complicated, I will be speaking with the same recruiter again next week for another similar position at the same university. I also have another interview lined up within the same university, but in a different department. This position has a pay range of $19 to $22 per hour, with the targeted hiring pay being on the lower end. Given my experience in administrative roles, I know I bring a lot to the table, but I’m feeling uncertain about how to negotiate my salary, especially since I haven't worked in higher education before. I'm trying to figure out the best way to approach this without underselling myself.