In that case, do you think doing an EMT-I would help get that sense of usefulness in the hospital (and also on the ambulance?)
Few hospitals have positions that will utilize an Intermediate's skill-set inside hospital walls. Even most EDs won't let medics do medic things while employed in the ED. Part of this has to do with actions from various nursing organizations to shut out other providers from doing tasks that they view as traditionally nurse jobs (start IVs, give meds, assess patients, etc). A medic is far cheaper than a nurse, and the fear is that hospital systems would preferentially hire more medics to do those jobs, rather than hire or retain additional nursing staff.
As far as getting a job as an EMT-B in the hospital, it can easily be done, you just would likely work as a CNA/Technical Assistant/Care Partner/Whatever-they-decide-to-call-the-lowly-assistant. I worked as one of those when I was a B, and it wasn't too bad (pay was MUCH better than the ambulance). However, it did serve to make me more bitter towards floor nurses, as I would do the vitals, clean up after, feed, record I/Os, put in/take out foleys and straight caths, do Q2 turns, etc about thirty patients (aka, the entire ward) each shift, while they would have their three or four patients only, and constantly bitch in the breakroom about how tough it was to work there, and how underappreciated they were.
Anyway, you could also look at various other tech jobs in the hospital, as the amount of additional training that would be required is less if you are already a B or I. A buddy of mine got a job as a phlebotomist while he was in Enhanced (Virginia's version of Intermediate-85) class.