In Tree Style Tabs, you can set the default behavior for new tabs. So ctrl-t will open a new tab as a child of the current tab. Or, as suggested, custom shortcut, or a separate new tab button for this.

+1

I finally switched from Firefox to Chrome today. Tree Style Tabs was the main reason I stuck with Firefox for the last couple of years. I installed Sidewise and realized it mostly does everything I need from Tree Style Tabs, *except* this. This is a pretty big one for sure. Though I imagine not too hard to implement.