to attain (reach, rise to) a new height (altitude)

https://en.wikipedia.org/wiki/High_ground

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Note added at 5 hrs (2022-03-02 10:03:18 GMT)
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In Sun Tzu's The Art of War, military leaders are advised to take high ground and let the enemy try to attack from a lower position.
https://en.wikipedia.org/wiki/High_ground

On October 16 the Chinese central defense force begin a major counter-attack to take the high grounds at Nanhuahua, during this battle the commander of the 9th Army Hao Mengling became the first Chinese army general to be killed in action during the Second Sino-Japanese War.
https://en.wikipedia.org/wiki/Battle_of_Xinkou

Lt. Col. Robertson’s 3rd Battalion was to the right of the 1st Battalion. The 2nd Battalion was to the right of the 1st Battalion. The 2nd Battalion was in reserve. Hill 362A was one of the major defensive lines for the Japanese. Its height of 362 feet commanded the last of the high grounds. The Japanese defenders were ordered not to fall back from this position, to die at their guns. The high ground advantage must not fall to the Marine attackers. And the Marines were instructed to take the high grounds regardless of the cost.
https://www.grunt.com/blogs/news/27th-marines-on-iwo-jima-st...

NEW DELHI : The Indian move to occupy the heights on the southern bank of the Pangong lake proved to be the game-changer in the entire conflict and ensuring that both sides disengaged from their forward locations.
https://www.livemint.com/news/india/gamechanger-idea-to-occu...

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When asked about the steps to ensure that China does not occupy the heights vacated by India, he said, “We will trust but we will verify and we have put our systems in place to make sure that there is no reoccupation of these heights. It is part of the agreement.”
https://thewire.in/security/disengagement-pangong-tso-win-wi...

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4 p.m. two Japanese-Manchurian detachments crossed the Soviet frontiers and attempted to occupy the heights 2 kilometres to the north of Lake Khasan
https://history.state.gov/historicaldocuments/frus1938v03/d4...

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Navy Yard and two and a half miles upstream at Benning's Bridge), and to occupy the heights above the Navy Yard and Washington Arsenal.
https://www.nps.gov/places/000/battery-ricketts.htm

you can't capture an unoccupied position (just exposed to fire)

standard army parlance.