Many of the suggestions given above are valid. I especially like the phaerimm idea and concur that godly involvement is necessary (since there are so many and they are quite involved in Faerunian events). Here's my not too thought-out take on your question (take it for what it's worth)...
The "greater evil" could be a displaced entity (divine?) from another world that is out for new territory. The Almighty Evil IT ravaged its native world, totally killing/enslaving the populace and having millennia to hone its craft of destruction. While scrying for other places to conqueor, what should IT come upon but the dear sweet and innocent Forgotten Realms. IT decides to invade and brings along IT's army of magic-immune constructs to aid in the assimilation of Faerun.
IT has a device (?) that yanks the FR gods from their homes and hurtles them into the Realms in avatar form much like Ao did during the Time of Troubles (except Ao is nowhere to be found/heard). <Consider this a good excuse to use F&P's avatar stats!
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> IT gives the gods time to pathetically scurry for corners to hide in amongst their most devout worshippers before sending out the constructs to reconointer. During this time, the evil gods cannot help but attack one another while within vulnerable avatars (again, much like the ToT). When the great heroes of the Realms (Elminster, et al) come rushing to aid the good, they naturally defeat the avatars just in time. But IT knows the history, has had thousands of years to prepare and when IT threw the gods to the ground, IT added another little surprise...when avatars are killed, they EXPLODE. And these aren't little explosions like before, oh no. We're talking nuclear detonations!! The kind that carry tainted magical remains across whole countries and mutate all it touches.
Once many of the popular heroes are killed, IT sends in the constructs to find the remaining avatars, thus squashing all hope of divine intervention. These constructs have numerous defenses and magical immunity as well as a ray to further weaken an avatar's powers. This is where such a post-apocalyptic campaign might begin. IT's constructs are everywhere (and they are of varying shapes and sizes) and the PC's may get to encounter a puny avatar that asks THEM for help! While a construct's primary directive is to destroy avatars, it also is directed to stamp out any and all power groups it comes across (good and evil).
Etc., etc. There's my half-baked idea.
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