The Katy Trail and OTET are two longer trails on your route to consider.
I don't know what time of year you expect to finish the GDMBR but if it were me, the other factor I would take into account is potential heat/humidity. The dew point map below is for July. Generally, about 65F I is listed as "muggy" and temperatures I start to find uncomfortable. So( the other item (somewhat at odds with the first) is I might consider swinging further north first. For example, roughly following the Rio Grande River north at least through Albuquerque and coming via eastern Colorado to connect into the Platte River valley and then across... However, I would be trading off some nice Katy Trail riding for slightly better odds of cooler temperatures.
In either case, I would come across the Erie Canal drainage towards the end.
Hopefully earlier parts of Route 66 are better but I mostly followed Adventure Cycling Route 66 between Springfield and Chicago ad found it not as pleasant as many other Adventure Cycling routes - with a fair amount of frontage road riding. The final parts of getting into Chicago are probably nicer though I actually followed mostly quieter/off-road parts from Joliet into Chicago and then a fair amount of trails across northern Indiana.
