Senior U.S. officials pressed Saudi leaders in a series of messages to quickly wrap up their air campaign in Yemen for fear of making matters worse, people familiar with the matter said, before Riyadh declared Tuesday it was ending the offensive.

Yet on Wednesday, Saudi airstrikes resumed in several parts of the country after Iranian-linked Houthi militants took over a military brigade in the southern city of Taiz, provincial security officials said. There was no sign of peace talks, though the Saudis had said they were shifting...