Hi,
Thank you for all that info.

The Kings Improvement was done in 1865 for the Model 1866 Winchester lever action rifle by a man named KING who worked for OLIVER F. WINCHESTER of WINCHESTER FIREARMS in NEW HAVEN CONNECTICUT at the time.
It refers to the loading gate on the side of the action as we are all familiar with on the model 94 Win levers which he was granted a patent for.
Prior to 1866 you loaded the old HENRY rifle from the muzzle end of the magazine by pushing the thumb slide all the way up and rotating the last several inches of the magazine tube to one side exposing the magazine opening.
Then you dropped the cartridges down the tube much like you would fill a 22 caliber repeater today if the magazine tube follower were removed.
With the KINGS IMPROVEMENT you simply pushed your cartridges in the loading gate on the side of the frame.

The 32-20 or 32WCF as they were stamped were not available until the MODEL 1873 lever rifle came out.
The models of 1866 and 1876 were no chambered in these rounds.
They were common in the models 1873 and 1892. However the serial number is XXXXX high for the Model 1873. Your rifle is a Model 1892 and it was made in 1913.
Grading:
Sporting Rifle; has plain straight-grip stock with capped forearm - Exc.: $5000, Vgood: $3500, Good: $1250, Fair: $700
Fancy Sporting Rifle; has checkered walnut, pistol-grip stock with checkered capped forearm - Exc.: $12000, Vgood: $7500, Good: $4000, Fair: $2500
Carbine; has 20" barrel with plain straight-grip walnut stock with one barrel band forearm - Exc.: $3500, Vgood: $2500, Good: $1700, Fair: $1200