Yes, he can.
The Scout rogue's Skirmisher feature says:
You can move up to half your speed as a reaction when an enemy ends its turn within 5 feet of you. This movement doesn’t provoke opportunity attacks.
Note that this ability uses your reaction, not your "movement". You don't have a pool of movement outside your turn, normally:
On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here.
There are no general rules for willingly moving (as opposed to being forcefully moved by someone or something else) outside your turn. You normally have a pool of movement on your turn equal to your speed, and none outside it.
The Scout rogue's Skirmisher feature gives them a special way to move outside their turn. It lets them use their reaction when an enemy ends its turn within 5 feet of them, and specifies that the Scout can move a distance up to half its speed. This doesn't depend at all on how much (or how little) you moved on your turn. You can use this reaction if the requirement is met, regardless of whether you moved on your own turn.