It works with straight copy for pretty much all drives except C
so if old letter was anything but C, it works since windows simply refuses to install critical things it needs to start on other drives.
So, go to drive E in explorer, check that you have all files shown in folder view options, no hidden files and show system files.
hit CTRL-A (or use mouse to select all, your choice)
Do not try to copy: (they don't like to be copied)
$RECYCLE.BIN
System Volume Information
Press CTRL-C (don't use X or cut, since you risk losing existing data if something goes wrong in the process)
go to drive H
press CTRL-V
Press WINDOWS-X
select disk management
change drive letter of E to something else
change drive letter of H to E
change drive letter of something else to whatever you want, possibly H?
Again, this works for other drives except C
If it doesn't work after reboot, all you need to do is to swap the drive letters back to reset what you did, it should work but.. if it doesn't, all you lost was some time taken by the copy process.