I assume you are going to be doing a fish-in cycle as the tank is larger than your existing one so it would be safer than leaving him in that? What temperature is the current tank (measured using an alcohol thermometer or a calibrated digital thermometer, not a liquid crystal thermometer)?
The best thing to do is probably move everything to the new tank, including fish, then very gradually add warmer water until the water in the tank reaches the minimum temperature on the heater, at which point, switch the heater on at the minimum temperature. Then, over the next few weeks, increase the temperature on the heater by 1 degree every 1-2 days until it is at what you want it to be. I recommend around 22 C for most tropical fish, including Bettas.
It is probably a good idea to add the water slowly anyway, because your tap water could be quite different in terms of parameters to your tank water as tank water parameters tend to drift over time (unless you do regular, large water changes) and fast changes in water hardness can harm fish.