I was like you. Freshman and sophomore year of undergrad, I kept getting rejected by the university hospitals to volunteer because the the vast amount of premeds who were trying to volunteer. I eventually did get a spot once covid hit and everyone had gone online and were doing their classes and stuff outside of town (my undergrad was a college town, and most students were not from this town).
However, that volunteering experience wasn't really the best. Minimal patient interaction, just getting them food, blankets, etc.
I'll tell you to instead go to private owned clinics (the smaller and less established, the better), and just go up to the reception and ask if they are looking for any volunteers. Have your resume, dress nicely, and let them know you can help out in any way. This was the best experience I had. I volunteered my time through many ways: reception work (checking in and checking out patient), assisting the nurse and getting the patients' vital signs as they waited to be seen by the physician, doing basic lab work, scribed, and more! I really got a well rounded experience getting a good insight into not only what the physician does, but also every other team member of the clinic. I was placed wherever they needed me most whenever I came into my shift.
So my advice is both. Keep applying to the hospital volunteering position, and look for other experiences too. Apply to privately owned clinics, hospice, free clinics (a lot of places of worship have free clinics and always looking for volunteers, I've seen a couple of food banks have free clinics too). You can also try to work on getting certified to work as an EMT, phlebotomist, MA, or a CNA and get clinical experiences those ways.