Hi there.
I found a solution for this issue.
First, I created a volume for each machine, used the same volume name, did it according to the docs on Fly.io.
Then I mounted the volume in my toml file, to make sure my media folder points to the volume, like this:
[mounts]
source = "media_data"
destination = "/code/media"
After that I just experimented a bit by adding the following code to my toml file as inspired by reading Fly documentation (there was already a similar section for statics, I just added one more):
[[statics]]
guest_path = '/code/media'
url_prefix = '/media/'
And it works pretty well! Now all images uploaded via admin interface are safely stored on the volume, and they are displayed by my website as well.
Not sure whether this is 100% safe usage of volumes so no guarantees.