I haven't walked the GR 11 but for other caminos in France you are spoilt for choice - Arles Way, Piemont, Vezelay ... and the list goes on. I like the CSJ site as a starting point as it gives good summaries of various camino routes.
In a survey the GR34 was voted the most beautiful or best hiking trail in France. But I don't know about the weather in October*. The weather might be much better in southwestern France where I have wanted to walk the Sentier de Cathare. Here are links to both.
This is the most famous of the Grande Randonnée long-distance footpaths! It’s a route steeped in history, salt and sea spray. This is the preferred long-distance footpath of the French: welcome to the Customs Officers’ path! It snakes along the coastline of Brittany for over 2,000 km, from...
The Cathar Trail GR®367 is an emblematic hiking itinerary. 250 km from the Mediterranean to the Pyrenees through the Aude Cathar Country and its citadels.
*Brief weather-in-France story...I arrived in Europe for the first time in Italy in September 1967. At a youth hostel in Florence I met a young girl of Austrian descent who lived in Belgium. I told her I was going to try to hitchhike to Paris and she asked me if she could accompany me as far as Munich. An American engineer picked us up near Bolzano in a brand new red Porsche. Here she was ascending the Alps in a sports car with two nice Americans. It was a sunny, beautiful day. She was, I am pretty sure, thrilled about how this hitchhiking idea had turned out. Oh yes, the weather. At one point, as the Porsche climbed upwards, she said to me, "Thomas, look at the sun. You're not going to see it again for 6 months." I studied in Paris until the May '68 protests when my school shut down. The young woman's quip about the weather held much truth.
When I walked the Chemin du Puy I recall a French saying - English translation. 'If you can see the Pyrenees - it will rain. If you can't see the Pyrenees - it's raining.' It sounds much better in French but you get the idea!
Thankfully while walking the Le Puy and later living in south west France, I found the weather to be just fine. And the rain when it came was really just a shower compared to the downpours I'm used to back / down here on the east coast of Australia.