I don't know if this is too late for your trip to JC, but other than the Civil War Museum, be sure to go to the capitol and see those Civil War exhibits. I heard they were having a special Civil War flags display but I don't know the timing on that. Fremont's house is in Jefferson City, but it is on private property. The state just moved Price's log cabin and are in the middle of restoring it. There is a National Cemetery which really isn't that large and it has a memorial to the Battle of Centralia. For some general history stuff, there is the 'Landing' where there are a couple of old buildings (right around the old railroad depot - Amtrak now) and there is just a small part of the entrance to the old Missouri River bridge left which shows the old art deco style. The Museum curator will be able to give a more detail description of what's there.
I really do not know about surrounding areas. From where are you coming? There are probably a few 'attractions' along the way such as the Battle of Moore's Mill, close to Fulton. I have not been there but it is probably just a marker.