Pour 1 cup of baking soda into the toilet's drain opening, if the toilet does not have more than 2 inches of standing water in it. Pour 1 cup of vinegar into the toilet's drain opening and wait until the solution stops fizzing and making noise, then flush the toilet again.
Place a funnel-cup plunger over the toilet's drain opening and press it firmly against the bowl. Pump the plunger's handle toward the drain opening and then away firmly while keeping the cup pressed firmly in place. Repeat this motion at least a dozen times.
Place the corkscrew end of a toilet snake into the toilet's bowl, taking care to not scratch the bowl with the end of the corkscrew. Feed the snake's flexible line into the toilet's drain opening and spin the snake clockwise with the handle.
Push the snake through any clogs you encounter in the toilet's drain line. If you cannot push through a clog, retract the snake 1 inch or so as you spin the handle counterclockwise, then spin the handle clockwise rapidly as you push the snake hard down the drain line again.
Turn the snake's handle counterclockwise and retrieve the snake once you have fed the entire flexible line into the toilet's drain. Clean any debris off the snake's corkscrew with a rag. Snake the toilet up to two more times if the clog remains.