William Seward didn't even get the nomination for President in 1860, but he was the natural leader of the nascent Republican Party at the time, and originally thought to be the leading choice. He was an excellent speaker, forceful advocate of liberty, and made an amazing Secretary of State.

One who worked with the great Henry Clay was Daniel Webster, member of the House, Senate, and also twice Secretary of State. A true statesman, the Senate named him one of the five greatest Senators of all time. He ran for the Whig nomination multiple times, but never received it.

Of the generals who sought the presidency and lost, probably my favorite was Winfield Scott, in 1852. Leader of the US Army in the Mexican War, after the Taylor/Fillmore administration, there wasn't enough support for the Whigs. The winner, Franklin Pierce, had also served as a general in the Mexican war, but simply was not the commander, as Scott had been. Scott continued to serve as commander of the US Army clear into the Civil War.