hedwards: Honestly, probably not, If the drive is making funny sounds, you're going to have to get as much of the data as possible the first go, there's no assurance that you're going to be able to recover sectors later on, it might even be asking too much just to get the files.
Not to scare you unnecessarily, but in a situation like this you're unlikely to have time to get every sector backed up.
Let's forgo the word "Backup" for "Evacuate", since I won't be using the drive I'm copying data from. I'm a stickler for semantics, especially since I have to deal with ambulance chasers whenever I'm not dealing with the ambulance passengers. Second, the drive was working just fine up until a
recommended firmware update (recommended by a Seagate tech support rep, so it's on them to foot the bill for the replacement), after which my BIOS reported that the drive was SMART OK but BAD. I exaggerated the "death chatter" bit, but I suspect that's the reason since it's running a little "louder" than normal. The actual "red flag" was a Reallocated Sectors Count failure.
Anyway, I'm want to try to lift all "loose" data (Videos, photos, music, etc) to an external drive, and then try to shift all program data to the new internal drive.