Starting to look like a cracking route 👍
I'll chip in with a few local possibilities that I've covered, although not on tour.
Connected with the Teversal Trail is a
network of others including the Silverhill and Five Pits trails, all pretty much interconnected and well worth exploring - Kudos to Derbyshire on this one. [There are a few barriers to negotiate as the trails cross roads]
The above trails take you almost into Chesterfield where you can pick up the
central TPT. The Chesterfield Canal provides about 10ish cracking miles towards Sheffield and is in good cycling order throughout. The onwards TPT into Sheff is also generally OK ... for a (sub)urban route.
Northwards from Sheffield to Penistone, the TPT is mostly good, offers some really interesting sections, and is worth putting up with one or two short, iffy sections. Onwards from Penistone, it can be a bit dodgy in winter (muddy) or after extended wet weather, but in summer is generally OK ... and up in the Pennines and onto the moors you get some great views.
A slightly dodgy, but short (and downhill) section drops you briefly on the A628, or you take the off-road but decidely MTB path. The section which follows, through the Longdendale Valley along the former Woodhead line makes that brief inconvenience worth it in my opinion.
That brings you to Glossop and out of my manor somewhat, but from there, there are some fine back roads to take you down to Buxton to pick up the Tissington/High Peak Trails. A slight but not too onerous detour would take you across to Marple to pick up the Peak Forest Canal at Marple to Bugsworth Basin then onwards Buxton via back roads. This section of towpath IIRC is also pretty good, though locals might be able to comment in more detail. It really is a lovely section though, in an elevated position with good views over the valleys.
Hope that helps a bit.