Daniel Webster stayed content(ish) in Congress, but what if he went bigger?
Bit of an understatement, if he stood in any more presidential election, he would be as famous as Harold Stassen or Ralph Nader.
He did want bigger. He stood in 1836, along with the other three Whig candidates, William Henry Harrison, Hugh L. White and Willie Person Mangum. But was only on the ballet for his home state of Massachusetts, due to the strategy of the Whig party in running several presidential candidates simultaneously.
In 1848, Webster sought the Whig Party's nomination for the President, but was fairly beaten by the military hero Zachary Taylor.
Even in 1852, he made his final campaign for the Presidency, again for the Whig nomination. He only gained support from New England and was a distant third behind General Winfield Scott, who received the nomination, and President Fillmore. The "Know-Nothings" put his name on the ballot without permission and he collected a few thousand votes, even though he died just before the election
But back to your question, in my honest opinion, the best year for having Webster run, is in the 1844 presidential election, following the refusal of incumbent Whig President, John Tyler, for being re-nominated.
If Webster, concentrates on speaking out on American nationalism, while avoiding the subjects of Slavery and Texas, he might just be able to beat Democrat James K. Polk, especially if Webster could keep Pennsylvania and New York, unlike Clay in OTL he would gain 62 from both their electoral votes. Taking the vote to 167 votes to the Whigs and 108 to Democrat Party.
As for Vice-President, he could go for:
Representatives from New York, Hamilton Fish
Representatives from New York, Millard Fillmore (OTL future President)
Representatives from Pennsylvania, John Sergeant
Representatives from Pennsylvania, Andrew Stewart
Representatives from Pennsylvania, Thomas M.T. McKennan
Webster's Cabinet
President: Daniel Webster 1845–1849
Vice President: Joseph Reed Ingersoll 1845–1849 [1]
Secretary of State: John C. Calhoun 1845–1849 [2]
Secretary of Treasury: Henry Clay 1845–1849 [3]
Secretary of War: Winfield Scott 1845–1849 [4]
Attorney General: Edward Bates 1845–1849 [5]
Postmaster General: William A. Graham 1845–1849 [6]
Secretary of the Navy: George Law 1845–1849
[1] Ingersoll, was not only Representatives from Pennsylvania, he is also famous as the son of Jared Ingersoll, who was a delegate of Pennsylvania to the Continental Congress and signer of the U.S. Constitution, while his older brother, Charles Jared Ingersoll, is a Democrat representative for Pennsylvania.
[2] Is already serving as Secretary of State, in Tyler's cabinet, I doubt Calhoun would turn down the opportunity to carry on.
[3] One of the foremost proponent of the American System, Henry as secretary of Treasury, I could see him fighting even harder for an increase in tariffs to foster industry in the United States, the use of federal funding to build and maintain infrastructure, and a strong national bank. I also picked him as I would love to see a cabinet meeting with Henry Clay, Daniel Webster and John C. Calhoun, who were all voted as three of the five greatest U.S. Senators.
[4] Scott, Commanding General of the U.S. Army, a strong image for the Whig party
[5] Bates, Former Representative from Missouri, studied the law with Rufus Easton and boarded with his family. Easton was Judge of the Louisiana Territory, the largest jurisdiction in U.S. history since the Louisiana Purchase. After being admitted to the bar, Bates worked as a partner with Easton.
[6] Graham, is an up and coming Whig politician, he had served as a member of the North Carolina House of Commons from Orange County, for seven years and serving twice as speaker, he was elected to the United States Senate to fill the vacancy caused by the resignation of Robert Strange, and served from November 25, 1840, to March 4, 1843.
[7] Law, a steamboat entrepreneur and financier, from New York, would want to help improve the navy insuring his own investments were safe.