That is a possibility, but a competitive shooter would have had a lot of other equipment, such as Shooting glove, shooting glasses, coat, lace-up boots, score book, spotting scope, shooting pad, etc. And, it doesn't seem that any of that was found with him. I have asked the investigator what make/model the rifle was, and what type of ammunition was found (military ball, match, or hunting). Do not have word back yet.
My feeling is that he either intended to do some target shooting, or that he was expecting a shoot out of some sort. It is also possible that he had simply recently purchased or obtained this rifle and ammunition, and was transporting it until he could either store it at his home, or do some target shooting. It would be nice to know what the sight settings were when the rifle was found, and if he had tools for adjusting sights, targets, target shooting equipment, or any suplies to clean and care for the rifle.
The National Matches at Camp Perry, Ohio always start in July and go through mid August, with rifle shooting being usually in August. You are correct, in that the 1903 Springfield was still the high power rifle favored for service rifle match shooting at that time. The M1 Garand was being used at matches in the early 1950's, but many more model 1903's were in civilian hands then.