It's weird, right?
So if we're not cheating, what you're saying is that as long as My Precious is in play and face-up, Gollum would be playable with the target company? I wouldn't call discarding My Precious a condition, I think it's done as a result so there aren't two manifestations of Gollum running around.
Another approach could be to look at the whole "play Gollum if" thing as an effect of the
agent attack, and not the agent himself. Then because the effect is something that happens after the attack has been resolved, I think for timing purposes you'd need to declare this effect as a part of the attack (declared when the attack itself is declared). Because the effect that allows you to play Gollum resolves with the attack itself, it does not get declared again after the attack. Instead the target company can now tap a character in order to play Gollum in the following chain of effects. In this scenario there would be no time to untap a character first.