"Stealth Bird" doesn't go face down unless you activate its effect to flip it face down (it's 100% optional). I've usually "Change of Heart"-ed "Stealth Bird", flipped it, and then left it face up so my opponent has to waste his turn to flip it face down.
But to answer your question, if you decide to flip "Stealth Bird" face down, destroying "Axe of Despair" in the process, you can then offer the "Stealth Bird" as a tribute to return "Axe of Despair" to the top of your deck.
- Andrew