What about cutting over from HWY 411 instead of going up to Kodak and coming down?
That saves about 8 miles off 66, but that is the section of 66 where you can make decent time. But because of the slower speed on 411, you aren't really going to save any time.
My daughter lives two exits down from Kodak and we have fallen in love with Cades Cove when the crowds aren't there. So we often go from I40 over to Cades Cove and then back up to Strawberry Plains. I go a bunch of different ways some of which I won't even begin to send a trailer down. As said, there isn't an easy way to get there.
It also depends on what time of day and which day you plan on getting there as that changes traffic on 66.