Fortunately, I had one of those "a-ha!" moments early one morning, and I could finally picture how to make the costume larger without it taking too much effort (or money). Unfortunately, I didn't think to take photos during the process, but the idea is so simple, hopefully words and after-photos will suffice.
Start by ripping out the side seam that extended from the bottom hem all the way up to the armpit, and down the arm to the hem at the wrist. Do this for both the seam that goes down the right arm, and the seam that goes down the left arm. If the cloak is lined (ours is), also rip the seam of the lining on both sides of the cloak.
Then, have the person who will be wearing the costume put the cloak on, so you can see how much fabric you will need to add. (In the photo below, the red arrows show that I added about 12 inches of fabric. My daughter wanted to make sure the cloak was really loose.)
(I was able to find fabric at the local fabric store that was a close-enough match to both the cloak fabric and the lining.) Cut 2 strips each of the main fabric and the lining. Then, with right sides together, pin one side of the lining fabric to the cloak's lining (the raw seam I had ripped). Sew that, then pin one side of the black fabric to the cloak (right sides together). Next, attach the other side of black cloak fabric to the other side of the cloak's raw seam that you ripped (still working on the same arm, though.) In the above photo, the arrows clarify the two lines of stitching that is needed for each arm. Then, turning the fabric inside-out, you can pin the final (for this arm) raw lining piece to the lining seam you ripped. Just make sure that the right sides are together. (After pinning and before sewing, I carefully turned the fabric right-side out again to make sure that I wasn't about to make a mistake.)
In the photo below, the arrows show the two seams that are on each piece of added lining.
Repeat for the other arm. All that is left then is finishing the hems--both at the bottom of the cloak, and at the sleeve. You could either do an overlock stitch on the outer fabric, or do a folded hem (make a narrow fold, then fold again). Either way, make sure your outer fabric hem covers the raw edge of the lining. Then stitch in place.
Of course, if you are skilled in casting the Engorgio spell, you might not need to do any sewing, but hopefully these directions helped any fellow Muggles out there!
Thanks for simple projects, and for inspiration.