The hospital bit was key for me both in getting started, and maybe more importantly for the transfer from admin that scripted to developer in engineering. They have a huge demand for people who can program, but the pay is "goodish" at best, and the culture is...slow and crufty. Lots of room for growth, and they are more willing to accept non traditional backgrounds. Best of luck.
If you keep running into an "employment experience" issue but you actually do know what you are doing, a dev I know used to be a maths teacher, created an LLC with a dumb name and created a consulting website. Then after a couple of years of no leads, simply listed themselves as an employee at said LLC on their resume. They are also one of the best devs I've ever had the pleasure of working with.