According to some online tutorials for sewing invisible zips, I had to iron out the curl near the zipper for best results (here's an excellent tutorial with lots of pictures). Other tutorials said that was unnecessary because that's what the invisible zip sewing foot is for -- to uncurl it while you sew so that you won't need to iron it beforehand. The part about the special function of the invisible zip sewing foot made some sense to me so off I went, WITHOUT any pre-ironing. Luckily everything turned out fine and here's the final result:
Now that I've tried it once, I think the ironing is necessary if you do not intend to buy additional sewing feet. You can then use the normal zipper foot which usually comes together with the sewing machine.