Camino Francés in September or summer?

I’m planning to do the full Camino Frances this year, and I have some flexibility in timing. I need to wait until after the beginning of June for work commitments. Then, the rest of the year I can start any time. I’ve been hearing that April is the best, but that’s not an option.

I’ve also heard that September is a great time, but I’m not sure if that’s more applicable to pilgrimages that start in the south.

I’m fine with crowds on the trail, but don’t want to compete for a bed. When would you recommend going? And how many days should I plan for at 10-15km/day, with a few stops along the way? I walk a lot and I’m in pretty good shape, but I want to do everything I can to avoid injury.