The Classic Pass is explained with a pivoting action of the lower half between the right second finger and thumb in both Royal Road to Card Magic and The Expert at the Card Table. I mention this because it is now a more common practice to execute the entire pass action using the left hand fingers without the aid of the right hand pivoting the lower packet. In short, the left/ lower hand does all of the work. However, while this more common technique may work for some people, allowing for both hands to contribute equally (using the pivoting action) is something that can increase the speed and by default increase the invisibility.
In essence, the left / lower hand should only be concerned with stepping the upper half of the deck to the side, not in it's full transposition. The right / upper hand then pivots the lower half of the deck around the upper half between the right second finger and thumb by pulling upwards against the left hand thumb. Refer to Royal Road of Card Magic for a clearer illustration.
For fear that I poorly explained myself here, I'll simply say that by refining the technique the speed of the Classic Pass can be increased.
Covering actions for the Classic Pass are personal preference, but I will share a quick practice tip that I use to better cover the pass by means of shielding with the right / upper hand. By tilting the front right corner downwards (assuming that you are right handed) the right hand will cover roughly half of the deck to a forward facing audience. This helps hide the main movement of the pass. However, keeping the right / upper hand fingers closed so that there are no windows nor breaks in the fingers, although it's not extremely difficult, can be made facil. I propose top palming three or four cards in practicing (my preference being four) the Classic Pass. This way the third and fourth fingers are forced to remain still with a natural contour that has no windows nor breaks between the fingers. If either the third or fourth finger move and create a window during the execution of the Classic Pass, the palmed cards will be dropped from the hand.
I hope that this was helpful and understandable.
-Cameron B.