Normally, if a tranny doesn't slip when it's cool it's due to an internally leaking pump or seal. As a result of the low pressure on that circuit it allows the tranny to slip, that causes friction and makes it over heat. Sometimes that can worsen the pump/seal leak.
Take it to a reputable shop that works on Toyota auto trannys so they can do some trouble shooting and diagnosis. There maybe test points where they can check to diagnose where the problem may be.
I've never worked on a Toyota auto but I worked on lots of US brands. I've seen T350's exhibit these sames symptoms and the fix was to replace an O ring in a shift solenoid that could be accessed externally, or replace the front pressure pump which was easy to do after dropping the tranny out.
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