During the 2016 campaign and his first year in the White House, Mr. Trump and Mr. Miller have developed an easy working approach to the president’s speeches, according to a former campaign adviser. Their process, which a White House official said began last fall and in earnest in mid-December for this speech, usually involves Mr. Trump telling Mr. Miller what direction he wants to take with the speech. He will then talk out some lines.

Then Mr. Miller, who drafts the material with a keen ear for applause lines he has heard during the president’s previous speeches, usually revamps the speech and hands it back to Mr. Trump.

On at least one occasion, the president has noticed that Mr. Miller struck a line he had said he wanted in. It is usually restored.

Mr. Trump took at least a few minutes out of his day for a photo op, posing in the Oval Office with the guests who are expected to be seated near the first lady, Melania Trump, during his address. In photographs released by the White House, Mr. Trump shook hands with some guests and flashed his trademark thumbs-up sign while standing next to others.

If tradition is any indication — and at least for today, it was — the last minute back-and-forth between Mr. Trump and his speechwriters could continue until the moment the president departed the White House for the Capitol.

Mr. Clinton was known to agonize over the details in his speeches until the last minute. He was also able to speak extemporaneously at times, including during the first minutes of his September 1993 address to a joint session of Congress on health care, when a speech from earlier that year was accidentally fed to a teleprompter.