The computer sees the removable device drive itself as a device and (rather than caring about whether there is media in it or not) assigns the same letter to it.
You are right in thinking that it would use the next available letter, which in your case would be the H: drive. If your netowork drives are mapped by a logon script, or batch file, then you should be able to right-click and disconnect the network H: drive, this would then enable you to use this (you might need to refresh the screen) as the USB drive that you've been trying to get to work on your machine.
As I said, it would be so much easier to get someone who has admin rights on the computer (whether this is as a Domain or local admin account) and change the letter through computer management....
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I've seen the light... It was green, flashy and attached to a Network Interface Card...
Whenever someone says "You can't miss it" I invariably do...