I ran across a reference to this illustration, entitled Jeff. Sees the Elephant in a catalog of William Reese Company, a high end New Haven, CT dealer of used books and printed materials. I hadn't seen it before, and it looks particularly interesting because it seems to prefigure Thomas Nast's later use of the elephant as a symbol of the Republican Party.
According to the catalog, the original was an 11 by 15 inch colored lithograph created by E.B. Kellogg and E.C. Kellogg in Hartford, CT circa 1861-1862. The description continues:
A . . . humorous political cartoon satirizing the Confederacy, and quite likely the first instance in which an elephant and a donkey were used to symbolize competing political entities in the United States. The Union is symbolized by a power elephant, who wears a blue coat and shoes and stockings decorated in the manner of the American flag. He carries the Constitution in his pocket and holds a sword in his right hand and eight cannons in his left. Behind him are more cannons, a pile of cannonballs, the flag, and the U.S. Capitol. The elephant stares at a donkey in the left side of the image. The donkey, dressed as a dandy and symbolizing Jefferson Davis, raises a monocle to peer at the elephant. He holds a plumed helmet decorated with a skull and crossbones. Behind the donkey stands an army of donkeys, carrying rakes, pitchforks, brooms, and scythes. A gallows in the background between the elephant and the donkey portends a bleak future for the Confederacy, The phrase "seeing the elephant" gained popularity during the Gold Rush and meant "seeing it all." In this instance, Jefferson Davis and the Confederacy see the fully array of Union might.
I don't vouch for it, but this site asserts that the elephant was first used as a symbol of the Republican party during the 1864 presidential campaign:
Despite a common assertion that the elephant first appeared representing the Republican Party in 1860, the first political cartoon using the elephant for the Republican Party was in 1864. The 1860 cartoon was actually just a shoe advertisement that happened to be on the back of a political magazine. During Abraham Lincoln's 1864 presidential campaign, a pro-Lincoln newspaper used the 1860 advertisement image to announce the news of a U.S. military victory over the Confederacy. Later that same year, the image was used to predict Lincoln's re-election based on the Republican Party's success in state elections.
Thomas Nast later popularized the connection beginning with his November 7, 1874 cartoon The Third-Term Panic:
So if all this is true, did "Jeff. Sees the Elephant" serve as the inspiration for the identification of the elephant with the Republicans?