The push for a fresh approach comes after Ukraine’s monthslong counteroffensive failed in its goal of retaking territory lost to the invading Russian army and after weeks of often tense encounters between top American officials and their Ukrainian counterparts.

The United States is stepping up the face-to-face military advice it provides to Ukraine, dispatching a three-star general to Kyiv to spend considerable time on the ground.

Administration officials argue that Mr. Putin is betting on diminished American support, pointing to his recent statements that if Ukraine runs out of NATO-provided ammunition, Russia would prevail in days.

They are searching for creative ways to keep Russia off balance with attacks against arms factories, weapons depots and train lines for moving munitions, and to score symbolic victories.

U.S. and Ukrainian officials said the victory was small but significant, a step toward cutting through Russian lines and pushing to the Black Sea, splitting the Kremlin’s supply routes.

“They have to fight smartly and efficiently,” said Michael Kofman, a senior fellow in the Russia and Eurasia program at the Carnegie Endowment for International Peace, who recently visited Ukraine.

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