TFS has an function called "doAttacking" that runs every 500 ms.
Then TFS also checks if you can attack when you move.
And lastly TFS checks if you can attack when you select a target.
This means, standing still, without spamming the red attack box on a target, the fastest you can attack would be 500ms. (twice a second).
If you move, you can attack each time you move, and if you spam the attack box (red box) you can attack faster than that.
There is no good way to get faster than 500ms attackspeed without source edits.
BUT, if you "must have" faster attackspeed and you don't want to edit sources you could make a new script in weapons.xml that repeats, and add it to all the weapons in your game.