Well, if 2015 is a valid reference (you never know...), it was good in late September, with sunny days. I walked with a t-shirt and sometimes a light jacket.
In the first weeks of October, the temperature dropped noticeably, and in O Cebreiro mountain pass, things got difficult, with wind, fog and rain. But after that, it was moderately cold, with sunny or slightly cloudy days, light rains from time to time, and colder nights. A medium range fleece was a quite convenient addition. And you have to give good attention to your rain gear.
As a picture is worth more than many words, I have inserted two.
Late September, early October, is one of my seasons of choice. Besides, it is not too crowded, apart from the last stages before Compostela.