I don't know of any way to do it using the Explorer icon.
What I did was to create a new toolbar, put it at the extreme left end of the taskbar, and then drag the normal icons as far left as possible so that only the toolbar title is visible with a ">>" beside it. With that set up, left-clicking on the ">>" shows a popup menu with folders in the toolbar that you can click on.
I tried this out, but I can't seem to get it next to my normal icons without squishing them. Another thing is, the toolbar I'm making is a folder with some folders in it, and when turn it into a toolbar, one of the folders is sorta showing. It's really awkward.
Ok, I got the first part down. No more awkward, barely peeking folder showing next to the >>. But how did you put the folder toolbar all the way to the far left? I tried unlocking the taskbar and then dragging my folder toolbar to the left, but my other icons are in the way...
I was confused about the little dotted separators and didn't know you had to hold them to drag them. Once I figured that out, it was a easy lol. Now I have a neat taskbar with a quick launch on the left side.