If you can access your files, you can copy your files. If the concern is that you only know how to connect from a full PC, consider plugging a laptop into the switch (or even just set up a VM).
Hard to give much more help without knowing your actual setup. But one nasty solution is to ssh into the server then connect to the running container (or mount the same storage into a different one) if there are some shenanigans going on there.
But yeah. My general rule of thumb is that if something needs to outlive the life of a container then it is being stored on the local filesystem or a zfs/ceph pool.