Is it appropriate as a referee to contact an author after the editor rejected the manuscript? This is related to this question, but it is more a special case of it.

I reviewed a manuscript, which was submitted to a reputable traditional journal. The journal has a single-blind review system. I liked the manuscript a lot, as it was closely related to my research interests. However, the authors did not do a great job in reporting the manuscript, and I provided a long review with several suggestions for improving the report. 

I suggested major revisions, but the editor rejected the paper. That is ok for me; it is the editor's responsibility to take such decisions. What is important is that the process is now closed.

By re-reading my review together with the other two referee's reviews, I came up with a couple other suggestions that I truly believe would benefit to the manuscript. Would it be a bad practice to contact the authors now that the paper is no longer under review? I do not mind unveiling my identity, and I would do that only because I want to advance the research in the paper. That is, I just want to help them.
One important consideration is whether the paper is publicly available (for example, being circulated on the web as a preprint).  If it is, then it's perfectly reasonable to get in touch with the authors and offer suggestions.  You don't need to address the question of how you heard about the paper if you don't want to.  If you do want to reveal yourself as a reviewer, I think it's fine, but I've run across people who disagree about this.

It's much trickier if the existence of the paper is itself confidential.  If for some reason I submitted a paper without circulating it publicly, and then after rejection a referee contacted me privately with detailed suggestions for improvement, I would feel uncomfortable.  It would look a little too much like an attempt to become involved in the research or writing, and I would wonder whether the referee might take offense if I just said "thanks for the suggestions" and didn't engage in further discussion or collaboration.

If you have confidential knowledge about a paper, then you must not even appear to be using it for your own benefit.  For example, you can't ask to start a collaboration or try to find out more about this research.

If there's no public preprint, then it's safest to make contact through the editor, and at the very least you should ask the editor about what you intend to do (the editor may well object, and in any case you need to avoid seeming secretive about it).  Even if the editor approves, you should be careful, since innocent actions may be misinterpreted.