Happy Birthday.
Odds are really good that the hard drive that's your gift is way bigger than the one you've got. Not to mention, judging by the looks of things, a lot younger too.
If that's true, and if it were me, I'd replace the old drive with the new drive entirely, and get the old one out of the machine.
What you describe can be made to work, but I think it leaves you with a less reliable machine, that you're not taking full advantage of.
Just my opinion. Best of luck.
rr