Yes, you can still use an auger. Slide it down the drain twisting as you enter it until it will not go anymore. When you get to this point shake it a bit and try to push harder. If it will not move, then you're likely at the blockage point. Now you either turn the handle on the auger, or if you have a cheaper snake, you'll have to twist it while pushing. The idea here is to break down the blockage and allow it to pass through.
If you use a plunger, note that at the top of the bathtub there is an overflow drain. Usually this drain is covered with a silver cover, or something similar. At the bottom of this silver cover is a hole for a second drain. (In case someone forgets the water running, it will drain out the second drain higher up on the bathtub instead of overflowing out onto the floor.) This hole needs to be covered and sealed tight, otherwise the air suction from the plunger will just be moving air in and out of this hole instead of affecting the drain blockage. Also ensure you have enough water in the bathtub for the plunger to create an air-tight seal.