Okay, perhaps "hide" is the incorrect word here: but during the War of the Last Alliance, when his armies were shattered and the defences of his kingdom destroyed, why didn't Sauron at some stage attempt to ferry the ring of power to safe and/or secret location, perhaps to the south or the east (where friendly-ish countries lay).
Instead, Barad-dûr lays under siege for seven years, and he eventually fights his last battle, in person, on the slopes of Mount Doom (notably the only location where the Ring was in serious jeopardy of being unmade). While he almost certainly would have lost considerable power being separated from the Ring, it seems that the writing was on the wall: there was no way he was escaping Mordor with his body intact. Why, then, leave his soul in the hands of his enemies?
After all, he couldn't guarantee that, if defeated, the ring wouldn't simply fall into the hands of someone strong willed enough to simply destroy it.