When you right click the Start Button and select Explorer, you are choosing to specifically explore the Start Button folder, not start Windows Explorer in a folder or drive of your choice.
If you access the E: drive frequently, your best bet is to create a Windows Explorer shortcut to that drive. Go to Start/All Programs/Accessories. Right click Windows Explorer and select the "Pin To Start Menu" option. This will place a shortcut to Windows Explorer on the left side of the Start Menu.
Click Start and then right click the new Windows Explorer shortcut and select Properties. In the Target Line, copy and paste the following command.
%SystemRoot%\explorer.exe /e, E:
Click Apply/OK.
Now, you will be able to click this new shortcut, once, to open Windows Explorer with the E: drive selected.