Your best single source of information on the M1903A4 snipers rifle is an article by Bill Hansen entitled "Snipers, Snarls and Snafus" published in the journal of the Remington Society of America in the Fall of 2004.

There is a link to Mr. Hansen's serial number table in the sticky section of this forum.
The same table appears in Vishooters website but without the very useful explanatory notes that are included in the Remington site.
The Remington Society of America (Remington 03, 03A3, 03-A4 Serial Numbers)
Vi Shooter's Web Pages - Extensive Information on M1903 & Variants - Military Surplus Collectors Forums


There are also links in the sticky section to TM9-270 which describes the operation of the Telescopic sight on the A4 and to TM9-1270 which is the 1944 update of the whole '03 family including the A4.
TM9-270 U.S. Rifle, Cal. .30, M1903a4 (Sniper's) Characteristics and Operation
TM 9-1270 U.S. Rifles, Cal .30, M1903, M1903A1, M1903A3, M1903A4 Sniper

This link will take you to the minutes of the ordnance committee which wrote the specifications for the A4.
http://www.nicolausassociates.com/PD...ber%201942.pdf

This should give you a good start.

Regards,

Jim