Gil's clairvoyance comes from his third NP, Sha Naqba Imuru, which he has in all his forms. It was first named in Fate/hollow ataraxia, and the full description of the NP can be found in child Gil's profile in Fate/Grand Order. Narita also talked about Archer Gil being a "clairvoyance user" in an issue of TYPE-MOON Ace, and related a conversation he had with Nasu about how Gil's clairvoyance would affect his memories when he was summoned in Fate/strange fake.

Sha Naqba Imuru doesn't appear on the status or weapons screens in the VN, but neither do a lot of the other NPs used in the story (Rho Aias, Enkidu, Nine Lives, most of Gil's treasures). If you reread the VN with Sha Naqba Imuru in mind, there are plenty of scenes where you'll notice him using it. For example, he was able to tell that Caster had faked her death, that Saber was no longer a virgin, that Berserker's true name was Heracles, that Shirou was preparing seventeen projections, and that Sakura was a Holy Grail, all just by looking at them.