I learned out of the Alfred Adult books, and in hindsight, I think the left hand was neglected in most of the pieces in those books. It was hard for me to play classical music at my level with all that left hand stuff going on!! If you are going to use supplemental material (every piano teacher I ever had would assign other music too) then maybe that won't matter. My current teacher says she does not generally use method books after about book 1-2 (beginner), but she says some students prefer method books because they can measure their progress. There is a TON of downloadable classical music for beginner and intermediate students, and somewhere on this website I think there is a list of easier classical music, and also the University of Iowa Pedagogy Project was a huge help to me when I was re-honing my skills after many years of not playing piano.
Topic: Best piano method books