Should be a date stamped on the barrel, top side, behind the front sight. behind a flaming bomb stamp, like NIGHTLORD40K has mentioned Would be a month-year. That crown DOES look a bit strange.
Yours is a double heat treated receiver rifle, and those were prized by the conversion gunsmiths because of their slick, smooth operation. They were very strong and safe. The nickel steel actions were often criticized for their "sticky" feel when manipulated. It is interesting to note that all 03s were proofed at the Armories at 70,000 PSI, even the early ones. We went through WWI with low numbered Springfields, for the most part.
The pics are greatly appreciated but we need more pics of the receiver, the bolt and the rear sight, especially the base. If it has a large scallop on each side it may be original to the gun. If not, it a Remington made WWII part. The bolt handle looks to be swept backward slightly, this would be incorrect , I think. I'm going from memory here. Does the bolt have an "R" anywhere on it. Or the letters N.S. stamped on the root of the bolt handle. If it has an "R" it is a Remington bolt and is not original to the gun. The NS stamp means Nickel Steel. If these things are present, the gun probably went through an arsenal rebuild. This does NOT detract from its value!! It is simply part of the rifles history. `
Just pulled out my copy of The Springfield 1903 Rifles , by William S. Brophy. The letters in a rectangle indicate a rebuild at one of the arsenals after WWI. AA is the Agusta Arsenal, but that "J" is a mystery. In the book, observed stamps are AAHI, AAL, AAHO, AAP, AAR and AAS. There is no mention of an "AAJ" And that "J" is clearly visible, it can't be anything else. However, Brophy does say, "These are the
known stamps."
1918 was a year of changes and improvements in many areas, The bolts went from a straight handle to slightly swept back. Nickel steel was introduced into manufacture. The stock went from one stock screw to two.
I'm guessing your rifle went through the rebuilding process at the Agusta Arsenal, which in no way distracts from the fact that it is worth four figures, easily.
CapnMac pretty much nailed it with his observations.