As TV crews and reporters camped out in the lobby of Trump Tower on Monday, recording VIP arrivals in hopes of detecting new members of the incoming Donald Trump administration, a middle-aged corporate executive drew little notice.

Greg Hayes, chief executive of United Technologies Corp., wasn’t there to see Mr. Trump. He rode the gold-colored elevators to see Vice President-elect Mike Pence, and the two men then shook hands on a deal that seemed impossible months earlier, say people familiar with the matter.

The...