[hs86]

(Not OP) Sidebery is half tab manager, half session manager. It stays in the sidebar, and if I collapse parts of my tree, I have set it to unload those folded tabs after 60 minutes. There is also an option to hide those folded tabs from the native tab bar.

Sidebery, Tree Style Tabs, and Tabs Outliner (for Chrome) all go beyond just adding a linear/flat vertical tab bar to your browser. They preserve a nested hierarchy for child tabs and allow you to restore the entire tree (or just parts of it) on another device, which is super handy if you often switch between desktop, laptop, etc.