At the start of your first turn of each combat, your walking speed increases by 10 feet, which lasts until the end of that turn. If you take the Attack action on that turn, you can make one additional weapon attack as part of that action. If that attack hits, the target takes an extra 1d8 damage of the weapon’s damage type.

 

It looks like "that" refers solely to if *specifically* the extra weapon attack hits? Or is it if any of your attacks hit you get an extra 1d8 damage? I would think it's the latter, but the wording makes it look like the former.

 Definitely the former.

If it was the latter it would use wording similar to the Zealot Barbarian's Divine Fury ability:

Divine Fury

[...] the first creature you hit [...] with a weapon attack takes extra damage equal to 1d[...]

Only the additional attack granted by dread ambusher does extra damage.