Problems opening tub drain

[t&krennovation]

We have what we think is a 'lift and turn' tub drain. We are trying to remove the stopper to clean a clog. We have tried twisting the plug counterclockwise, but the top does not come off. There is a screw in the top of the plug, we removed this as well, but it still does not come off. There is no set-screw on the underside of the stopper that we are aware of. Any help would be appreciated. Here is a picture of our drain:

[LazyPup]

Under the stopper there is a tube connected to the stopper which surrounds a pin that is screwed into the threaded center hole one the tub drain cross. The tubular section slides up and down on the pin allowing the lift and turn action. When new the screw in the top holds the stopper on the end of the tube but as they age often the tube becomes corroded and nearly permanently locked into the stopper making removal very difficult. In some instances you can lift the stopper as high as you can then reach in with a pair of small needle nose vise grips and grip the pin to unscrew it from the drain cross however, even if you get the stopper out, it is very unlikely that you will be able to clear your clog from the tub drain.

Most clogs occur either at the Tee where the overflow meets the tub shoe drain or in the trap.

To snake the line remove the overflow cover and insert your snake in the overflow riser tube which will then permit the snake to go through the tee and on down to the trap.

I am attaching an illustration that shows the layout of the Waste & overflow piping. The illustration shows a lift type stopper in the overflow pipe. You will not have that so you need not worry about removing the lift type stopper, but you would snake your drain in the same manner as illustrated.