Update: clicking on an obstructed tab is actually possible. But I wouldn't see what I am clicking.
I might want to activate or just close that tab.
When I close multiple tabs I don't need anymore, and am moving the mouse up and down the tabs list, a lot of nervous flickering goes on with the tooltips. I don't think vertical-tabs should fix the flickering, but if you can make the tooltips not obstruct other tabs in the process, that would already be a huge improvement!

I'm not the author but I made some research and figured out the following things. In short: it's not possible for CSS. It's possible for JS though but it requires a lot of tricky code that still will never work smooth.

It's not possible for CSS since Atom creates a global tooltip that's placed in the root DOM element and has nothing related to tabs. It's controlled with API inside of the tabs package (built-in features are packages as well). What about JS, that tabs package stores an internal variable called hasBeenMousedOver in every tab. It's false by default, and when it's false, atom renders a tooltip and sets it to true. I know how to render tooltips myself but I can do nothing to that variable, it's impossible to set it to true manually (packages are isolated from each other), I ended up my attempts by having two tooltips displaying at the same time: mine (on the left/right side) and default (below) ones.
It's also possible to just catch existing tooltips and change their position but sometimes they display instantly (without any delay) so custom code just can't keep up with tooltips and they display below a tab. That happens once per new "fast cursor entered" tab but that's still can't be considered as a good solution.