There is a Wikipedia article which is very extensive that lists methods of divination. There are three main suffixes I can discern from these
-mancy // From Latin mantia or Greek manteia, meaning either prophesy or divination
-oscopy // From Greek skopein (to look at)
-spicy. // From Latin specere (to look at)
haruspicy a form of divination by natural phenomena, especially from
inspection of the entrails of animal sacrifices.
hepatoscopy (a particular form of haruspicy by observing the liver)
extispicy (another term for haruspicy)
auspicy (another term for augury)
orniscopy (another term for ornithomancy and augury) the practice of
reading omens from the actions of birds
hieroscopy (another term for hieromancy)
horoscopy Divination by celestial bodies
anemoscopy (another term for aeromancy) Divination by behaviour of
ceraunoscopy or brontoscopy Divination by thunder and lighting
astroscopy (same as astromancy or astrology) Divination by stars
It seems to me that the original terms have been expanded in popular fantasy novels, video games and films. For example a pyromancer practices pyromancy, which is divination by fire. However in popular fiction a pyromancer would usually be a mage/wizard/warlock/witch who can summon fire and hurl fireballs at you. An aeromancer, who practices aeromancy (divination by actions of the wind) might be able to cast attack spells that blow their opponents away or encase them in ice.
But I'll just add, you'll unlikely find either -oscopy or -spicy satisfactory if you want to describe a magic user, because, unlike the suffix -mancy, these do not have the connotations given to -mancy by probably centuries of use in fiction to describe sorcerers.