My recollection on trying various methods of speed shooting is that having the hammer all the way forward against the firing pin (or the "firing pin carrier" in many of the center fire block designs) often causes resistance to opening the lever. I find a Spencer lever already sticky enough without the added resistance. I have never fired a Spencer using rimfire ammunition, which might make a difference.
From what I have read, there was no official federal manual of arms for Spencers. In Marcot's Spencer Repeating Firearms "First R&R Edition" at pp162-64 is a photocopy of Captain Barber's Manual of Arms for the Spencer. Captain Barber commanded the 5th Independent Company of Ohio Sharpshooters (which, along with its fellow independent companies was famously armed with Spencer rifles) and his manual deals specifically with rifles, although most of it could be applied to carbines. I do not see any reference to use of the half-cock. The firing sequence involves a "Load" command that involves turning slightly to the right so that the shoulders are in the correct alignment for aiming the rifle, then "briskly" working the lever while holding the rifle at something close to our modern "port arms," or diagonally across the chest. From that ready position the soldier can be commanded to shoulder arms (i.e. to resume marching but with a loaded rifle) or to continue the firing sequence. If firing, the next command is "Ready!" at which point the soldier cocks the rifle, followed by the command, "Aim!" when the soldier puts the rifle to his shoulder and aims, followed by the command, "Fire!" After firing, the next command is "Recover," which causes the soldier to return to the "ready" position--rifle (with expended cartridge case) held diagonally across chest but still turned slightly to the right with the left shoulder towards the target. The manual does not talk about repeated firing, but the "ready" position is correct for executing the "Load!" command, so the command sequence, "Load, Ready, Aim, Fire" would make sense to fire a second shot.
Another source, Spencer, The First Effective and Widely Used Repeating Rifle * * * by John C. McQueen, reproduces the contents of a July 12, 1864, letter from Colonel Charles C. Walcutt to the Ohio Veteran Volunteers stating that he had obtained Spencer rifles for them and then describing the manual of arms to be followed and various other Spencer-specific matters. He describes the manual of arms only in terms of how it differs from Hardee's manual (with which I am only vaguely familiar), but I find no reference to placing the Spencer at half-cock during the loading and firing processes.
A cautionary note: In both Barber and Walcutt the Spencer reloading process for filling the magazine (not the "Load" command in the firing sequence) involves inverting the rifle and placing the muzzle on the left foot. Although that reloading method is historically accurate, I suspect anyone adhering to that level of accuracy might get a stage or match DQ.
--DJ