I was mistaken. Alito's concurrence actually goes a bit further.
As I understand it (I'm not a lawyer, I don't play one on TV, and I didn't sleep at a Holiday Inn Express last night), the majority opinion was somewhat narrowly tailored to include the trespass as a component of the search. In his closing remark, Scalia said that if it had occurred without the trespass, they might still decide the same way, but don't have to worry about it right now.
Alito's concurrence, joined by Ginsburg, Kagen and Breyer, was considerably more vague. They discounted the role of the trespass, but considered it a breach of one's reasonable expectation to privacy, and therefore a search.
Sotomayer filed a separate concurrence, which supports both explanations, and suggests that they need to go quite a bit further in establishing privacy rights in the digital age.