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...