After completing the game, face the following hidden bosses by performing the corresponding tasks:
Hecate(Shakugan no Shana) Speak with Enomoto (person from Iriya no Sora; the real version of the fake version you fought in Chapter 1) in Dengeki Academy (found somewhere in the hallways where Dokuro-chan is found). Go to Iriya no Sora 2 (Book #9 in your bookshelves where you can revisit the worlds you completed before). Then, go inside the building where the colored elevators are found. Take the yellow elevator and she'll be there.
Accelerator(Toaru Majutsu no Index) Speak with Shizu (person from Kino's Journey; the man with a dog next to him, the real version of the fake version you fought in Chapter 2) in Dengeki Academy (located outside of the Dengeki Academy. Note: You may need to go out one more time in order to trigger him if you did not see him somewhere outside Dengeki Academy). Then, go to Kino's Journey 1 (Book #2 in your bookshelves where you can revisit the worlds you completed before). Then, go to the place where you fought Shizu and you will see the closed gate behind Shizu from before is open. Enter that gate and he will be there.
Aleister Crowley(Toaru Majutsu no Index) After you defeat Accelerator, in Dengeki Academy, go to the place where you purchase items and Dengeki Cards. Once you enter that room, a cinematic will be triggered. Aleister will be there and he will challenge you.