It depends where you are on the trail. Basically, if you are going to walk for 6-7 weeks you are going to see a shift from one season to the next -- even if it's only the whisper of the next one in the trees.
This year I finished in Santiago on Sept. 9th. In 2014, I arrived in Santiago on October 9th. This year it was a transition from a blistering summer to a warm fall with cool mornings. In 2014, when we were walking into O Cebreiro there were wee flurries in the dark morning, but it was the height of summer by the time we were back to more regular elevations. Take light weight gloves, layers (merino base, a fleece, and a Buff). You'll be fine. Merino is wonderful because it stays cool in the heat and warm in the cold. I had one long sleeve and one T for each trip. I had a T for sleeping in too, and would generally wear that while doing laundry. I was able to check into a spot about once a week where I could have my laundry done (with friends to make it more affordable), and wait in a fluffy robe provided by the hotel or pensione, etc. Everything was as perfectly clean as one can expect on Camino, and I was never too hot or too cold.
The first time I did not take gloves and had to buy some in León and they were not great. This time I took light-weight running gloves and pulled them into service for most mornings in the mountains at the end.
PS -- I discovered my favourite fleece (my ONLY fleece) at the Harrican Shop in Astorga in 2014 -- made by the UK company "Trespass" -- it's stretchy instead of rigid, and is darted at to provide a nicer fit on a woman, and it has "thumbies" to help keep hands warm. My first one is now rather stained from constant used with poles so I bought a second one for when I want to look more "put together" and was able to find it on Amazon. The stretchiness makes it wonderful as it avoids pulling and does not chafe at the seams.
PPS -- IMO yes to a rain jacket; just get one that is light-weight and wear something under it with long sleeves to prevent dampness. I took one this time and I was so much less grumpy on the one rainy day than when I trudged through driving rain in Galicia in a poncho.