Geez, I ain't no trans guy anymore either, this may require my using brain cells that ain't seen no blood in about 20 years !
Some questions;
Is this a transbrake body ?
Full manual "reverse" pattern shift ?
There are some issues with these in the valve body, but, if not, then lets continue;
"Reverse" involves the application of fluid to the Rear band servo actuator, and the front/high gear clutch Pak if memory serves me correctly. The pressure to both, is "bumped" from normal "line" pressure when reverse is selected, through the valve body, so as, to compensate for running the helical cut gears backwards(more parasitic resistence).
Anyways, If it is shifting crisp/fine, when hot, on the 2-3 upshift, we can assume High/forward clutch integrity for now.
However, the rear/low band servo is probably not "patent" when "Hot" causing the "hot" condition you mentioned. The reason it works fine when cold is the thicker fluid viscosity at the lower temp.
To try and diagnose the above, test for SLOW, "manual" low/1st gear application as well when "HOT".
In other words, try shifting directly from "neutral to first" when hot, and see if "it's slow" there too, or, if it feels like a "dual engagement", IN, and then IN again, as opposed to going from neutral to 3rd/drive.
If it does, start looking at the rear servo, adjusting the band won't last because the problem is in the sealing mech on the rear servo Piston.
Dunno if I remember all this correctly, ain't been in a 727 for years, got "too easy" once I found a good tranny guy, and didn't have to do my own anymore !
FC out.