September and October are uncertain times for weather in Memphis, especially the last several years. You'll have heat (mid to upper 90F) and humidity (mid 60s to 90s RH) to deal with until the 'switch' anywhere from late September to late October. Hopefully, you'll get to see some nice fall foliage change before you leave but I doubt it. It's been dry in the region and that mutes the colors.
That region is nice but you'll be right on the edge of 'good' and 'bad' areas for interesting sites. Travel East and North into Kentucky if you get the chance. Forget heading West into Arkansas (except for Northern Arkansas) or NorthWest into the bootheel of Missouri. South into Mississippi gets more … interesting … the further South you go but not necessarily in a good way. If you like to gamble, then Tupelo, Mississippi is close.