WASHINGTON—A billionaire Russian-Azerbaijani family’s efforts to push members of Donald Trump’s inner circle to ease human-rights sanctions against Russia during and after the 2016 election were more extensive than previously known, according to thousands of pages of testimony released by a Senate committee on Wednesday.

Rob Goldstone, a British publicist working for the Agalarov family, on multiple occasions contacted associates of the Trump campaign and the Trump Organization after Mr. Trump launched his presidential campaign...