We're always surprised how many veterans of the noble trade of plumbing don't know the best way to use a toilet plunger.
Instead of pushing down on the plunger like it's an old fashioned butter churn, the best way to use the tool is to gently compress it against the bottom of the toilet and then pull up.
What happens when you do this is that you pull the offending clog back up into the bowl, so the water can run out freely. Sorry, but we don't have a best way to then deal with the offending clog once it's in the bowl. That's up to you.