Typically our team will start off the same way you have described. Sometimes it is worth looking for exact same controller board revisions (just for tolerances)
Then, if we are sticking with sensible cost options, try EnCase, if you have access to it. Failing that, try the stick it in a plastic bag in the freezer for a couple of hours method - this works way more than you might think. We have used it for a couple of global banks within the last few months.
Beyond that you start getting into serious money options, which end up at scanning the entire disk with an electron microscope (many hundreds of thousands of pounds!)