If you’re not quite ready for an interview, ask someone at the museum for an informational interview. See if you can take someone in the department you’re interested in to coffee in order to hear about their job and give you advice. People love to talk about themselves and their job, and it’s likely they would love to feel important enough to share their story and advice with you. You will learn a lot about the many paths people take to work in arts institutions, and you will almost definitely gain a connection. (And, although you should try to pay for the coffees, they will probably treat you, because they were once a broke college student too.)

One thing that is super important about the informational interview: DO NOT try to weasel a job out of it. Seriously. They know you’re looking to break into the museum world–everyone understands the underlying reason for informational interviews–you don’t need to put it out there. Be subtle by not mentioning it at all, graciously thank them for their time, and there’s a good chance they will say something along the lines of “feel free to contact me if you have any other questions, and I’d be happy to keep you in mind for any internships if they come to my attention.” If they don’t offer something like that, don’t bug them to do so–it puts them in an awkward position if they didn’t really connect with you.