Pull drive out. Grab some bags of desicant (comes with a lot of electronics now adays, those packets with the little beads in them), put drive and dessicant into a ziplock bag (or 2), then place drive in the freezer for 20-30 mins max (less time needed for laptop/2.5" harddrives). You don't want the drive 0F (don't want drive dripping wet w/ condensation), just chilled down enough

Pull drive out of freezer, plug into a working clean computer, and image the drive with whatever tool you want to use. You shouldn't hear the clicking noise, at least initially. Once the drive warms up, you may get the issue again.

I've done this several times for clients who have harddrives that are clicking on start up (initial power sent to drive, you hear drive motor start up, then keep hearing a repeitive clicking, which is the drive head trying to find Track 0 on the drive and slapping against the head limiters). If your symptom is not clicking, then this will probably not work.

I would only do this if using an official harddrive recovery company is out of the question (not in budget, etc).