When does Finn add his result? Is it immediately following the die roll or is it during the modify results step?

During the relevant modify dice step. As all modifications do unless the card concerned specifically says otherwise.

To elaborate on @thespaceinvader 's answer: The defined timing windows are on page 18-19 of the Rules Reference (1.0.3). Finn uses the timing of while which means the ability triggers at the default part of a multi-step process such as an attack.

Adding a result, as Finn does, is one of the defined types of dice modification listed on page 9 and so should be done during the appropriate part of the attack.

For comparison, look at Heavy Laser Canon which says, "After the Modify Attack Dice step... " Or Crackshot, "... before the Neutralize Results step..." Which both give a specific timing outside of the default time for dice modification.