I think he wanted the job - or he wanted to be inside Hogwarts somehow. Some of the most important objects he'd ever touched were there, artefacts of sorts.

[ Because there is no way Hermione is going to tell Remus all the things she knows about Horcruxes - especially not after the books she forced herself to read. ]

You were an excellent teacher, Remus, I promise. Both at Hogwarts and in the Drabwurld.