It is a relief the weather has turned cooler and there is a bit of rain. Unless however it is prolonged heavy rain it should have no effect on the need to water trees planted / transplanted in the past five years (or plants in pots).
Minimal rain will only touch the very surface of the soil and this is often consumed by grass and not tree roots.
The primary benefit of cooler weather and bits of rain, is that a tree will transpire less. Quickly however a moisture deficit in the soil will occur (as now) and watering has to be increased.