Are you interested in history?  Food?  Cultural attractions?  So many great options in that area which would make this trip more exciting than "the usual sightseeing."  

Have you been to New Orleans?

Bits of the Trace are scenic. Between Nashville and Tupelo is nice and can detour to Shiloh civil war national park. The area between Tupelo and Jackson has a few nice areas too.

But as others have said hwy 61 is full of history, music, etc and is a great route.

I recommend Tishomingo State Park in MS and just off the Trace. Excellent hiking and scenery:

https://www.mdwfp.com/parks-destinations/state-parks/tishomingo/

We drove a large part of the Natchez Parkway this spring. And we were a bit disappointed, because the scenery was not so interesting. Skyline drive war more interesting. The best was that there were no trucks on the road. :)

Here is a website with all campgrounds near the parkway:

https://www.nps.gov/natr/planyourvisit/campgrounds-on-or-adjacent-to-the-parkway.htm

And here is a website were you can find accommodation:

https://www.nps.gov/natr/planyourvisit/upload/Accommodations-Adjacent-to-the-Natchez-Trace-Parkway-02-15.pdf

If you are especially interested in the history of the Trace and enjoy stopping at each and every historic marker to read them and get an understanding of the route, then fine. However, as already said above, one of the reasons you'll fine little on this forum about the Trace in details and any comments on the best stretches is because most of us road trippers who have done the many scenic highways and byways of the USA have not found the Trace that rewarding for scenery, viewpoints and especially memorable highlights.

I don't regret driving a few stretches of the Natchez Trace, a gentle, verdant road with no commercial traffic is a very pleasant break from driving I-40, but it was far from a highlight of a trip covering much of the route you describe, and when I suggest to friends where to visit for those who are interested in that part of the south, I now realise I've never ever mentioned that road.

As Eric notes, if you found doing all of BRP and Skyline boring, driving the Trace won't be better. There are none of the elevated viewpoints between the trees and apart from near the Nashville end, no waterfalls that I'm aware of, for example.

Vicksburg and Natchez are both interesting destinations, and smaller and easier to manage than either Memphis and Nashville, but there are several smaller towns we can suggest if you have time for leisurely exploration, assuming that in general your interests include history, music (including music history and live performance of various genres) and of course food.

You don't mention New Orleans. Is there a reason for that? If you want to avoid big cities, then that's your choice, but it's a great city. The only thing to mention given your likely dates is that 'in normal times' from Mardi Gras to early June, NOLA has some sort of event on almost every weekend, bringing crowds and higher hotel prices to the city.

SWT

Of course, highway 61 is these days just a regular highway, but it links up more appealing cities and smaller communities and there are the opportunities here and there to get views over the Mississippi from some bluffs.

To the trip itself:

What's a more specific timeline for this trip rather than the vague 'extended trip'? Just so we can help you gauge distances versus time and points of interest for the trip?

What's a more specific time of year than spring/summer? While nowhere mentioned really gets much of a winter, the interior of Tennessee can get frosty mornings with the odd icy bit of road at times. Possibly something to consider if you spend most of your time in coastal areas.

Jackson Falls, Fall Hollow and Owens Creek are waterfalls that can be easily visited on the Trace.

As we drove it in spring, there were many miles of damage of a hurricane north of Ross R Barnett Reservoir. All trails were closed, and as I see now Cypress Swamp still is closed.

My daughter's comment on the Natchez Trace as a 5 year old...."Dad, this road is like a big green tunnel!"

Thanks for all the great information. I enjoy U.S. history so waypoints and historical sites are of great interest as well as short hikes/trails to scenic spots. I think some misread my OP regarding BRP/SD where I said that we found some sections to be boring; but what I didn't say was that other sections to be magnificent scenery and quick side trips to small towns and markets.

Thanks again for all the tips as this is what makes TA such a great resource.

Stay well