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Does such vent orifice prevent the tank from reaching atmospheric pressure? Is the hole really very tiny? This does not make a lot of sense to me... looks like even the tiniest hole would bring the fuel tank to normal pressure quite fast.
No, as long as there is liquid phase in the tank the pressure will be the supply pressure (if still connected to the supply tank) or the pressure as defined in the N2O phase diagram. The vent hole is very small and typically you keep the fill going as the tank is venting.
 
And you will fly the rocket immediately after the fill is finished?
When the liquid N2O starts coming out of the vent the motor is ready to light. I start ignition after this happens but keep the fill going. I stop the fill after motor ignition.
 
And you will fly the rocket immediately after the fill is finished?
Here is a video of a flight. You can hear the start of the fill and the hiss of the gaseous N2O venting. Then the sound changes (obvious, gets louder) when liquid starts to vent.

 
I typically maintain the fill tank at between 650-850 psi which generally achieved by keeping the tank temperature between 15-25 C. As Troy suggested above, to keep the temperature down in Summer, I use a reflective blanket (e.g., car windscreen sun cover). To raise the temperature in Winter I will leave it in the sun for a while to get the pressure up into the above range.
I place my tank in a 5 gallon bucket and drape it with a wet towel to keep it cool on the warm days. Even on cool days the sun will warm the tank which is not good.
 

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