This will likely take an hour or two, and you will have to watch that the bath temperature stays above the planned developing temperature, and you may have to heat the water to do the task. Then use the Tai technique with developer preheated to the right temperature. Do not prewet the film with warm water. Almost certainly you will have to keep the tank and the additional developer in the heating water until development is complete. Home development of Minox color film, even something as forgiving as the new process C-41 films, should probably be something reserved for when you need pictures right now.
Temperature control is simpler the closer the temperature required is to room temperature.
In the first place many film and chemical manufacturers can provide a chart of developing times for different temperatures. Then it is only necessary to take note of the temperature of the developer after it has been poured in and to monitor temperature shifts during the first few minutes of development. You can estimate the 'average' temperature and develop for the time appropriate to that temperature. To sum up The Minox developing tank is an extremely elegant way to process Minox negatives, but processing today's films requires more care and thought than 'souping' older films in standard types of developer. The tank is so convenient that I have even carried it, along with small vials of film developer and fix, on trips, processing film each night to ensure that I could avoid exposing it to too many airport X-ray machines.