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As mentioned before, most of these rifles were issued to Volunteer Guard and US Infantry troops for the Spanish American War. The Springfield Research Service books don't have your specific serial number listed. The two closest I found: 368445 and 368540 were both issued to the 3rd Texas Volunteer Guard in 1898 ( Company I and A respectively). This is one of our ( Texas National Guard) units. They were mobilized for the Spanish American war and trained but did not go overseas.
Texas Military Forces Museum
Thank you so much for your help I greatly appreciate it.