How to make thin corn tortillas
Water temperature is one of the most important things and is hardly even mentioned. It should be medium.
Thinner dough can be achieved with adding touches of warm water. The more malleable the dough, almost to being sticky, the thinner you can get them. Most people use Maza Harina. I use Masteca flour.
Don't bother with a press. Use the advice above: two plastic bags (gallon size). Form the dough first into a golf ball size. Put round dough onto a plastic bag and press it out with your fingers into a round shape with a ridge around the edge.
Place other plastic bag on top so the dough is between the two bags.
Get a pizza stone, a pizza oven slider, or a book. Press down; pound it going out from the center.
Meanwhile I use two pans which are heated to medium/high. Lift the book (or whatever) and check the tortilla. Should be beautiful and round.
Peel off the top plastic bag, take the other one with the tortilla still stuck to it, and slap the tortilla into the first hot pan. It should peel off of plastic and lay there. I work the tortilla from one pan to the other; flipping sides so both get done. Then I flip it back to the first side in the first pan. After a moment tamp down on the tortilla. Its center should rise a bit, fluff up. It is done.
Nothing like what you buy in a store.