If you try to create a radio station based on Evanescence on Pandora, you might notice it really doesn’t come close to picking bands like them very well. It usually suggest songs by the following bands: Avril Lavigne, Breaking Benjamin, Plumb, Flyleaf and Linkin Park.
First off you need to add Within Temptation, Lullacry, Leaves’ Eyes and The Divine Madness they are the bands that comes closest to sounding like them. Lacuna Coil is a little more rockish, but you’ll want to add them as well. Naio Ssaion has the same pop goth metal sound as Evanescence, but with electric violins. Apocalyptica is mostly a instrumental band that uses cellos, although they have a few songs with a male or female vocalist.
Two bands you may or may not want to add are Nightwish and Epica. Nightwish to me sounds like 80s speed metal with modern distortion and medieval singing. Epica has the medieval vocals as well, but the guitars are a little milder. But it just depends on your taste, I’d check them out some samples before you add them though (Nightwish on Pandora, Epica on Pandora).