Instead of free admission, maybe there could just be more advanced free historical/museum content sent to your phone or to your house via a low monthly fee or a pay-what-you-want kind of thing.

People pay artists on a service called Patreon in order to access new songs, funny photos, and exclusive videos. Some bands even have a monthly gift they send to their Patreon followers (a post card, a small toy, etc). It's all very personal and makes people feel like they're a part of the community of fans. They're getting an insider look at the musician's life, and I think this could totally work to pull people into behind the scenes museum processes.

I don't know what the song or poem equivalent would be in the museum world, but this kind of personal, and insider content could get people to pay something extra to feel more involved with the cultural institutions in their neighborhoods.