The gun used multiple propelling charges placed along the barrel's length and timed to fire as soon as the projectile passed them by, to provide an additional boost.
My idea is to use multiple smaller combustion chambers which run co-axial to the barrel with multiple high surface area ports. Each of these combustion chambers would be fueled with a stoichometric mixture of oxygen-hydrogen or oxygen-methane. As the projectile passes, the combustion gasses ignite the fresh fuel in each chamber and provides more pressure (throughout the barrel) and higher temperature. Assuming there is no projectile blow-by (and with the ammunition I'm using, there wouldn't be) there would be no issue with timing. Some of the gas from each combustion chamber may be lost but I wouldn't be concerned given the large flammability range of hydrogen.
Any thoughts? Before I'm told this isn't practical, I know. It's research, and an experimental launcher.
Tl;dr: using staged combustion chambers for moar pressure and temperature throughout the length of the barrel.