As someone who presses open, I know a lot of the frustrations for side pressers trying to change. I keep thinking that maybe next year's Bonnie Hunter project I will try pressing to the side as she gives very good directions on which side. I'm more comfortable open and haven't yet taken the learning opportunity.
There are a lot of little things that add up to big differences:
I keep my chains of pieces together until pressing, being able to set up and tug on groups makes a big difference to get the seam to stand up. I'll press maybe 4-6 and then cut them off, even if I have 12 units on the ironing board and more on the floor. Then I'll tug the chain forward.
Direction matters! When you are pressing triangles and have the choice (not so much in HST), put the blunt end towards your iron and press towards the point.
When you have more complex seams and have the option, try to put them on the uphill side and avoid going over them with the iron. For example, the Shaded 4-patch, where you have a solid triangle joining a pieced unit, put the solid triangle down and the pieced triangle up.
When you are joining blocks, it is better to lift up to reposition than to slide along.
I don't use starch, but my fabric is very crisp from pressing it as yardage, another help. High humidity is not your friend.