It depends where you spend the month of June and it depends where you are from (i.e. what weather you are used to). Daytime temperatures may be anything from 10-15 to 30°C and the worst, I find, is cold rain.

So bring a few layers and if it's really that cold, just wear them all together and stop for coffee once in a while. I have rain pants and a rain suit, and one extra possibility is to wear these (pit zips open) if and when it's really cold. I also carry neoprene kayaking gloves which I find great when it's +5 or +10°C and raining: my hands stay warm and relatively dry.