Homemade Corn Tortillas

I've made corn tortillas at home now a few times with the usual Masa flour / water mix. I have a press which works pretty well.

They always turn out ok, but just ok. For tacos, etc., they don't have that elasticity / flexibility that you get with good store-bought corn tortillas, and so they don't hold up very well when you use them for tacos. Also I can't ever get them as thin as the store bought ones - eventually the press just destroys the tortilla or the dough gets too thin and I can't peel it from the plastic wrap.

Usually I agree that the homemade stuff beats the store bought stuff every time, but in this case I am not so sure.

And I might be willing to accept that homemade / traditional corn tortillas are just different and that is what makes them so great, except I have been to good mexican restaurants who make their own tortillas and they are more like the store-bought ones than the ones I make at home, and so that leads me to believe that I am missing something.

Anyone have any tips on how to get thinner, better homemade corn tortillas?