You need to look at the clamps that hold real rockets on the launch pad until they motors are burning evenly. The space shuttle uses (used a similar system) them too.
You could design a system that holds the rocket against the base and also holds the cork in until the clamps are released. Of course, be aware that you are creating a pipe bomb and it will be somewhat dangerous.
And this is why, today, NASA is going broke so fast!

Ladies and gentlemen, let me introduce to you the latest and greatest development to come at us from the great minds at NASA:

Those smartie pants also tried to pass us a small cylindrical piece of wood with a small hole drilled through the sides of it as a form of cork-stopper, but we're too smart for them and sent it back because we caught it in time!
Long gone are the days where kids earned their model rocketry chops with the kerosene can in the shed and uncle Fred's old box of grenades.