As I quoted before, Dice Rolling actions may be targeted earlier in the same chain of effects. Cards like New Friendship are played in response to the declaration of the influence attempt, because the dice rolling action occurs upon resolution.
The only exception to this is when the dice roll is an active condition of an action, e.g. the One Ring's strike cancelling ability.
In the case of Good Sense Revolts:
Playable on an untapped agent.
This is the active condition that must be met upon declaration and resolution.
Tap the agent who may then make an influence attempt against an ally, faction, or character.
This is the effect of the card. The wording makes it clear that the influence check occurs at the time of tapping.
MELE Rules wrote:A short-event's effects are implemented; then, it is discarded.
A card's effects are declared in the same chain of effects as play of the card itself, as I've always understood it.
Otherwise, it would be impossible to fizzle hazards with Deeper Shadow because all card play would resolve before all card effects resolve in a chain. Since Good Sense Revolts does not indicate it lasts until the end of the turn, the Agent may only make the influence attempt at the time Good Sense Revolts resolves.