When the Saints Come Marching In:
Rather than have more gods, why not have multiple denominations? Christianity has managed to splinter itself quite nicely, as has Islam, Buddhism, Hinduism, etc. All espouse a sort-of core system, yet you can have Mormons and Jehovah's witnesses alongside ELCA Lutherans and Gnostics.
To use Artemis, from the previous example, In Greece she was an (officially) asexual goddess of the elites, but when morphed into Diana in Rome, she was considerably more tolerant of commoners and at least some of the belief systems clearly suggested she was a Lesbian.
Further, you can have more sub-sets by introducing saints. These semi-divine beings can represent aspects of the angels (gods) they work under, and they can have their own worship practices, temples and dedicated shrines, infinite numbers of new backstories for clerics that love that stuff, and so on. Saints can be specific to each domain, since they are derived from mortals from those domains, thereby achieving local character(pardon the pun). With saints, I've even let my clerical characters create their OWN saints to follow, so they can structure their own practices the way that's most fun for them to play. There's nothing more frustrating than when you have a clear vision about how you want to play a cleric, and then the DM decides to flip the rules around and say you can't do it (admittedly, the DM that rule swapped did it to please his girlfriend, but you get the point).
This is not to say this approach can't create drama. When a denomination shifts far enough to stray into a new alignment, the resulting schism can be violent. Guess what? The same great drama that makes good storytelling makes great campaigns. You can add all sorts of fun role playing, where two churches to the same god come into conflict, or need to negotiate to achieve peace, or argue over the possession of a given artifact (or even the significance of the artifact...)