This post contains spoilers from The Diplomat season 3 episode 4 from this point forward.
In the previous episode of The Diplomat season 3, Kate and Hal worked overtime to make sure Penn wouldn't follow through with her consideration to bail on Hal as vice president. With the nomination secured and Hal sworn in, episode 4 opens with Kate going for a run to let off steam with a security team running behind her. Back at Winfield, Stuart warns the team about Kate's demeanor. When she returns, she's in a much better mood than any of them were expecting.
Stuart's surprised that Kate isn't stressed or frustrated, but she's taking it all in stride and sees her position as Second Lady as an upside. No one can turn down a meeting with her. She wants to meet with the person at the Foreign Office who deals with resource extraction for oil from water. Kate has a whole plan of using her position for good and getting as much accomplished as she can. Eidra arrives for her meeting and Stuart proposes Eidra running a new project to save her job.
Eidra isn't impressed with the offers and she doesn't see the light at the end of the tunnel. While at the Foreign Office, Kate bristles at her security team's proximity since she still views herself as the ambassador rather than the Second Lady. Before her meeting, she's swept into a photo op with Trowbridge, which turns into the meeting about drilling licenses. Trowbridge throws a wrench in the discussion and insists on calling Hal. Kate leaves with him and misses the meeting with Dennison.

Eidra can't find a way out of the Roylin mess
Eidra pleads her case about the Roylin situation, and she's asked if Trowbridge asked for the CIA to kill Roylin. She refutes that claim outright. Trowbridge wants Eidra fired, but Dennison doesn't want her to lose her job over this. They're at a crossroads about how to save Eidra's job. Kate speaks with Ganon about the drilling licenses, and he agrees with Trowbridge to bring the conversation to the White House. Ganon misunderstands Hal's intentions and fears he will be fired.
Needless to say, Kate's annoyed with how often people are using her as a means to contact the vice president. Stuart tells her she should probably just call Hal, and she tried to. But she couldn't even get through to Nora, and Hal hasn't returned her call. She puts on her sparkly black dress to prepare for a cocktail party and connects with Eidra, who doesn't think there's a way out. Eidra also gets brutally honest with Kate that doing two jobs can only be done poorly, even by her.

Kate and Dennison's relationship heats up
During the cocktail party, Kate speaks to Trowbridge about Eidra and asks him to offer her some leniency. It takes a bit of pressing, but he quickly accepts in exchange for Hal agreeing about the North Sea. She quickly tries to get Hal on the phone, but she still can't actually speak with him. Hal steps out of his meeting to call Hal when she sends him their emergency word: Arden. In Dennison's office, she checks out his personal photographs while she waits for Hal's call.
Hal worried that something horrible happened to Kate since she sent their emergency word, but she only needs him to comply with what she needs to make it through the cocktail party. Taking this call presents Hal with some real problems. After ending her argument with Hal, Kate calls on Dennison for a quick escape without running into Trowbridge. Kate opens up to Dennison about being secretly separated from Hal, but they still have an agreement to be together publicly.
She uses the opportunity to also make the slightest move on Dennison. It's awkward at first, but he pulls her into a kiss. Before long, the two are making out in his office and they both admit they have wanted to have sex for a while. They're interrupted by a call and return to the party. Later that night, Kate calls Hal, but he ignores her call. When she has to ask her Secret Security detail to unzip her dress, it's clear that she's feeling lonely without Hal, as much as she doesn't like him.