What you will need to do is to pull the toilet and run the snake from the underneith side up into the toilet. Many times something gets lodged down into the toilet trap and you have to push it backwards out the bowl, the way it came in. Since the trap of a toilet is larger at the bowl and smaller at the outlet, things can become lodged inside. I have had to do this many times. Once you clear the toilet from the bottom side and have the toilet off the floor, test the drain with a bucket of water to make sure there is nothing in the toilet arm underneith the floor.
The reset the toilet with a new bowl wax and closet bolts.