The coefficient of discharge is really a factor that is used to determine the discharge of water from an outlet based on the velocity pressure measured at middle of the water stream, or from the highest velocity in the stream. Over 10 psi the turbulance on the hydrant steamer outlet is too high to simply factor, so a pitot reading doesn't work. The variables from hydrant to hydrant are even more significant.
The turbulance from a steamer outlet can be high for several reasons, two of which are that the water flow has just turned 90 degrees and the steamer outlet is close in size to the barrel (here the steamer outlets are 4-1/2" and I think that the hydrant barrels are 5-1/4" inside.)
Think of water flowing in a river where it makes a 90 degree bend. Fast flowing water would be at the far side of the bend, and at the inside corner the water might be barely moving. If you took measurements of the water flow across the river shortly after the bend, you would expect the water near the far side to be moving faster than water closer to the near side. The flow in the middle of the river would be somewhere in between the inside and outside measured flow. The difference between the inside and outside is greater as the water velocity increases.
If you need higher water flow to get a good flow curve from a hydrant, you need to have a way to accurately determine the gallons flowing from the outlet. A properly installed flow meter would work. A straight piece of pipe could work. I suppose you could attach a large pipe with multiple (stream straightened) outlets. If you measure the stream from all the outlets, you would commulatively get an accurate measurement of flowing gallons per minute.