I've been to that museum several times and even got a behind the scenes tour. Anyone interested in that type of thing should consider joining The Company of Military Historians.
The nearby statue of Sgt. York on capitol hill (just up the street from the museum) has him holding an '03. There is also a war museum at the foot of the capitol building. Don't forget to see that too. Fort Nashborough (recreation of the pallisade fort) is also within walking distance. And if you're in Nashville, Fort Negley is within driving distance. It was part of the original chain of forts built by the Union to defend Nashville from the Confederates. Don't forget that Franklin Battlefield is nearby as is Murfreesboro (Stones River). No shortage of Civil War history around there.