ADDIS ABABA, Ethiopia — More than a year into his tenure as the nation’s chief diplomat, Secretary of State Rex W. Tillerson has a set of diplomatic tools that he and aides use at nearly every stop.

There’s the tight smile Mr. Tillerson wears when asked about insulting remarks by President Trump that have angered or bewildered entire continents.

There’s the shrug or dismissive laugh when he is trolled or overshadowed by Russia or China, rivals the Trump administration has yet to develop a coherent strategy to contain or surmount.

And there’s the signature pause that follows when his aides are asked why he has yet again failed to schedule a meeting during his trip with opposition or rival political leaders — the kind of encounters past secretaries of state routinely undertook to emphasize the importance of pluralism, and to hedge against sudden changes in government.