Hardening has set in
Sounds like the seals inside the tranny have hardened. My wife's 63 Comet had that in the Merc-O-Matic tranny. Unit was about 20 years old when this was happening. She would start to drive, and it would not shift out of low until the fluid got nice and hot to soften the necessary seals to hold pressure and shift into Hi.
A rebuild fixed the problem.
HammerWerf
1999 k2500 Suburban Pyro, Tranny Temp, Boost, K&N Air Filter, +3.5 qt tranny pan. 190000 miles
1984 C20 249k miles DSG gear, SM-465, 3.73
1983 C30 CC dualie. 308K miles on Chassis, TH400, 4-45 A/C Looking for a TurboCharger