Many obstructions in a bathtub drain can be dislodged manually through the overflow plate.
The first thing to do without removing your tub stopper is to remove the tub overflow plate. This is easy enough to do with just a screwdriver. Now you can use a simple plastic drain snake that can be purchased at any hardware store and has little notches down the side that grab hair and debris to easily pull out of the drain. If you do not have a plastic drain snake, you can take a wire coat hanger and bend the end into a little hook to reach in and pull out any stuck hair.
Pull out as much hair and foreign material as you can reach, and then flush the drain thoroughly with hot water.
