Your mention of a state park reminded me of Trail of Tears state park north of Cape Girardeau. It marks the spot that the Native Americans crossed the Mississippi River on the Trail of Tears march to OK. I think it's worth a visit, and it does include a scenic overlook of the river.
Big Springs State Park near Van Buren is another good stop, as someone mentioned earlier, although neither of these state parks would take that long to tour. Both do have some walking trails.
I would also second the recommendation to see Elephant Rocks State Park, if you have children with you, because they really enjoy the trail and climbing the rocks. It's located more to the north and is in the Arcadia Valley, which will give you some beautiful scenery, as well.
You can find more information about the different state parks at:
This has a map of the different regions, so you can look at parks in the southern MO area.
If you decide to go west and stay on 160 to Springfield, and have children with you, I think it's worth it to stop at Mansfield and tour the Laura Ingalls Wilder home. You can see Pa's fiddle, as well as other Little House on the Prairie memorabilia.
Northern Arkansas is beautiful, especially the further west you go, so you may want to consider dropping down into AR also.