This is an excellent question! I was intrigued enough to do some digging, and here is what I found.
From Sears biography of McClellan on Google Books - after being commander in chief of all Union armies, there simply was no where to put McClellan after his final dismissal in November 1862. Burnside, Hooker, Pope, etc. commanded only the Army of the Potomac. McClellan had (however briefly) been commander of all Union armies.
Pope was sent to command armies in the Dakota War, Burnside got his corps back, Hooker got a corps. None of these were nearly as harsh a "demotion" as any position McClellan could have been offered would have seemed.
Furthermore, and more speculatively on my part, Burnside and Hooker both offered to resign after what could only be understood as catastrophic defeats. Pope was relieved of command after what again could only be seen as a catastrophic defeat.
Antietam was a mixed bag. It could be argued to have been a Union victory. McClellan had not offered to resign - in fact, did not see any reason why he shouldn't retain command. He was being fired, and he knew it. He could not graciously save face by accepting demotion. The only course to save face was the stiff upper lip. As indicated in his letter to his wife, "Of course I was much surprised; but as I read the order in the presence of Gen. Buckingham I am sure that not the slightest expression of feeling was visible on my face, which he watched closely. They shall not have that triumph. They have made a great mistake. Alas for my poor country!" (quoted in Landscape Turned Red). If he held his head high and marched off out of the military, victim of the evil scheming Republicans, he would still be the hero of the story.
And this maneuver catapulted him to Democratic nominee for President.
In short - Lincoln had no desire to offer it, Stanton couldn't think of anywhere to put McClellan that wasn't an insult, and McClellan probably wouldn't have accepted it anyways.
From Sears's biography:
https://www.google.com/books/editio...dq=biography+of+mcclellan&printsec=frontcover