President Tsai Ing-wen of Taiwan met with Senator Ted Cruz of Texas in Houston and then flew off to visit leaders in Latin America, seeking to promote her island’s cause as it gets positive signals from President-elect Donald J. Trump.

The meeting on Sunday with Mr. Cruz, who lost to Mr. Trump in the Republican presidential primary contest, suggested that Ms. Tsai was looking to expand her ties to the Republican Party as it takes control of the White House and keeps its grip on Congress.