How to snake a bathtub with no overflow drain?
Your tub has an overflow drain.  You just have to find it. Remove the two screws next to the combination drain toggle &amp; overflow plate.  Now gently pull up to reveal the overflow hole and the drain plug:



Get out your snake (you've got a snake, right?) and thread it down the newly revealed hole.  Also remove the single screw holding the drain screen. Use some sort of semi-flexible plastic rod to clean out the horizontal section of the drain (a suitable tool comes with some bottles of drain cleaner, else you can improvise).

Don't bother with drain cleaner -- it is not right for your situation.