Vertically docking Tools toolbar to left of map and right of TOC in ArcMap 10.x?

There's a thread at the ESRI Forums on the issue:

Currently, the toolbar cannot dock between the Table of Contents and map. The closest thing is that the tools can be docked vertically to the left of the TOC.

For now, unless someone codes a very clever workaround, you can contact ESRI support for updates on the issue; refer to bug number NIM050280.