Lightweight puffy-style jacket

A very lightweight rain jacket should be all you'll need--more for rain than temperature.

Hi, I love my Patagonia Merino Air crew https://www.patagonia.com/product/womens-capilene-air-crew/36525.html?dwvar_36525_color=VJOG&cgid=root&isSearch=true#q=merino%2Bair%2Bcrew&lang=en_US&start=1

In June you probably don't need a puffy jacket. Just a couple of layers and a raincoat will keep you warm. And you can always buy (or bring) a scarf and you'll be fine. I like the merino air crew because it's very smooshable, not gonna wrinkle, the black or grey look classy, and it's so very comfortable. You'll love having it at home too!

Mean low temperatures in June will be low double digit C and higher (50F plus). Layer with a rain/wind (uninsulated) shell, something long sleeved but light, and a light wicking basellayer. A light sweater to nip the evening chill should be all you need for June at lower elevations. I'm assuming that you trekking the French Alps is not on your travel map.

Paris
https://www.yr.no/place/France/%C3%8Ele-de-France/Paris/statistics.html
June
17.5°C mean normal
21.8°C mean high

13.3°C mean low
8 days precip per month

I would have a polartek for cool evenings, or warm sweater, or lightweight down vest (I have the cheap very lightweight ones from Uniglo) that you can layer under the rain coat. It is cold and rainy in Paris now in early May and that has been sufficient for me under a lightweight trench coat. In June it is more likely to be hot than cold, but you want to be prepared and especially for cool evenings on say a Seine cruise. You shouldn't need a down jacket. If it were oddly and unexpectedly cold you could go buy one at Uniglo in Paris.

That kind of jacket will be too much. Take a light weight fleece instead. Layering is your friend.

Go for the puffy shirt. It will make you look like a pirate.

@FastEddie, thanks for the laugh!