How to override 3rd gear not shifting when cold

Hey, I’m currently dealing with the same issue on my 3.4L 5VZ-FE Tacoma Prerunner.

I’m from Suriname and have been driving this truck for over five years without major problems, until recently when I replaced the water pump and removed the thermostat.

The issue started about a month ago. At first, I only noticed it occasionally during cold morning starts: the shift from 2nd to 3rd gear would take unusually long. I figured it was time for a transmission service.

I changed the transmission fluid and also replaced the transmission filter, which still looked relatively clean.
Unfortunately, the problem persisted. I posted here at the time to ask for help and advice.

About two weeks later, I noticed a sudden spike in engine temperature — the temperature gauge shot up unexpectedly.
I then flushed the radiator and replaced the coolant, which seemed to solve the temperature issue.

However, just three days later while driving to work (and still experiencing the shifting issue), the water pump failed. Water started leaking from one of the holes behind the water pump pulley.

That same day, I replaced the water pump and also removed the thermostat completely, allowing the cooling system to flow freely.

Since then, I've noticed that every morning I have to wait 5 to 10 minutes until the engine reaches a certain temperature. Only then does the transmission shift normally.
If I drive off immediately, the transmission gets stuck in 2nd gear — and only shifts properly once the engine warms up.
I usually drive pretty fast on a daily basis.