If you have something to probe with, you may want to investigate your prospective drift to make sure that it isn't covering a big tree stump, boulder, or clump of brush. You must plan your cave so that there will be at least one foot of snow overhead. This provides enough structural strength and insulation. If you have the option, keep the entrance downwind and sheltered. Also, if the drift is big enough, start the entranceway below where you anticipate the floor of the cave to be.
Warm air rises.
Therefore, if the door is below you, the air heated with your body will be less likely to escape. Dig into the cave and push the snow out behind you. A partner, if available, can help dispose of the removed snow. If possible, remove the snow in blocks or chunks and use them to build a snow wall around your entranceway. This will be very helpful if you have a small emergency cave with little room.