I wouldn't have thought Puglia would be too rainy or cool to go exploring then. Depends on you to some extent, obviously!
I've been in Lecce in Oct / Nov and the weather was perfectly OK for exploring. In fact to be honest up to the middle - end of Oct it was still really quite warm. Warm enough for me to get sunburnt... (although I do have v.pale skin!). Early Nov I think it rained a bit but not much, and it didn't really get what I call cold until December. So I reckon you'll be fine!
Amalfi's likely to be the same I would think - I've been to Sorrento in Oct and it rained once but generally it was quite warm. ..
For the past four years I have been to Puglia in October and November and I can honestly say it is a crap shoot with the weather. One year it was beautiful spring weather and the next it was chilly and rainy. I agree that it would be a lovely time to discover the area. Since it should be nice and cool for you, it will give you an opportunity to indulge in some of the local foods that are a bit to heavy to eat in the overwhelming heat of the summer.
http://www.weatherbase.com will give you historical weather averages.