SINGAPORE—A tariff agreement with Vietnam emphasizes one of the White House’s top priorities in this frantic round of dealmaking over global trade: Slamming shut any back door routes for Chinese goods to enter the U.

With the dust far from having settled on this week’s US-Vietnam trade agreement, one thing is clear: companies operating in what’s become one of the most vibrant Asian economies are on notice to move up the value chain.

Pressure for trade deals is mounting as the temporary reduction of Trump's reciprocal tariffs is set to end in a matter of days.