As the age unfolds, it becomes clear that Melkor's own second in command, Sauron, is becoming a new threat. However, for thousands of years, Gil-galad doesn't personally get involved in the fighting much as the Second Age plays out. Nevertheless, he remains close on the edge of the story.
For instance, when Sauron arrives in the area disguised as Annatar, the Lord of Gifts, Gil-galad and Elrond are among the few who recognize that the mysterious stranger is a threat. Later, when Sauron starts a war over the Rings of Power, Gil-galad sends an army, commanded by Elrond, to help. He's also given two of the Three Elven Rings, which he eventually passes along to new keepers.
Most of the time that Gil-galad comes up in the text, though, it's with the Númenóreans. He becomes good friends with the Númenórean king Tar-Aldarion early in the age. Toward the end of the age, after Númenor is destroyed and its survivors set up the kingdoms of Gondor and Arnor, Gil-galad becomes besties with their leader, Elendil. He even builds Elendil some towers in his new northern kingdom as a kind of welcoming gift.
Throughout all this activity, Gil-galad remains on the periphery of the story. However, as the age comes to a close, Sauron reappears in Mordor and begins attacking the Númenórean exiles, sparking the events known as the War of the Last Alliance.