I press open and I pin a lot... every leading edge and about every 1-2 inches, usually a minimum of 2 pins, but typically 3 per piece (start middle and end). I use long quilting pins and the tops are well below the seam line or the feed dogs.
I'm looking from the narrow top down and lining up the seams and seam allowances. Some shapes don't quite work how you might think because of we are thinking in terms of the final size and not the seam allowances... So your HST might seem a little off on the first seam without the other sides.
I keep the pieces together in their strings and to help line up the seams for easier pressing, I'm usually pressing sets of 4-ish at a time (depending on size). First press from the back. You want the more simple pieces to go under your iron, to avoid snagging and other problems. Then flip over, give a quick little pat on the front and let it cool in place while you work on the next set.