Something I have come across a few times is a reference to the Dry Canteen Trail. This is/was a Boy Scout "trail" that takes Alexander McCook's I Corps path from Mackville to the battlefield. I've been doing a bit of searching, and was able to get some great info from Perryville resident Sam Reid. First, the old map.
How about a patch for completing the trail? Scouts had to walk the trail with empty canteens and were not allowed to stop by locals' homes to ask for water, in an effort to recreate and understand the hardships that McCook's men went through during the drought ridden summer of 1862. Of course, being a hiker and backpacker, this about one of the worst things one can do (hike without water), so encouraging scouts to not drink water is not a wise decision. Hopefully they have lifted that restriction a bit!
Here is the route of the Dry Canteen Trail from Mackville to the park. Just a bit under ten miles, mostly on roads.
I am thinking that for future walking tours at Perryville I might offer this walk as the Friday warm up to the Saturday main event. It will involve some car shuttling, but it does offer a) exercise and b) a chance to interpret a bit more of October 8th to those that may be interested.