No, that's not how it should be. It should go from the transmission, to the radiator cooler then to the axillary cooler. That is how they are from the factory if they come with one and that is how an aftermarket one should be hooked up. Transmission fluid should be kept below 250 degress, I don't like to see anything above 230, 300 is cooking it.
I got the temperature incorrect for proper operating temperature. So thank you for the correction there. I remembered 300 being a bad number but that was about it without re-researching the subject.
As far as the auxiliary cooler, if it has OEM equipment from the factory, it is indeed setup correctly for that vehicle.
Older vehicles that don’t use transmission temperatures as part of the equation on how the engine operates don’t care if the transmission is over cooled.
For vehicles that do monitor the transmission temperature and do not come with an auxiliary cooler, running the auxiliary cooler after the cooler in the radiator can cause problems. Transmission fluid that is too cool will force the engine into warm up mode or keep it there, throwing off fuel trims, emissions, and can cause the engine to run poorly. It was a thing we saw a lot in the HondaSUV forum and that I read about other places with other vehicles. Running the auxiliary cooler before the radiator cooler prevents that.
Of course, if the auxiliary cooler is too big, no matter how you route it, there is going to be problems on vehicles that measure transmission temperature for it operating parameters.
Of course, the best way to make sure the transmission isn’t over cooled is to install a thermostat.