Later on, I went back to the hospital as part of the Butcher's Bill mission, and again attempted to leave through the back door. This time however, the door was wide open but the bars had been put back in place. I was not able to interact with the bars, and therefore couldn't exit through the door.
Not a major issue as I was able to take the long way out back through the hospital, but still a frustrating inconvenience. Thought it might be worth a mention in case it's something that can be looked into!