You'll be grand
Don't be afraid to ask if you can join people head out in the mornings at the evening meals the night before or as you walk along. Failing that find out when people are leaving in the mornings and leave at the same time. Most people are more than accommodating and before you know it you will have your own little camino family
Clothes wise bring enough light layers for cooler/windier/wetter weather and you should be ok. I brought 1 Icebreaker tshirt and 2 long sleeve Icebreakers plus a fleece hoodie and rain jacket as my upper layers - that way I could just wear a tshirt in hot weather or layer up in cold weather or when hiking early morning. If you use hiking poles consider bringing gloves (or use spare socks!) for cooler mornings but again you may not need them for this particular stretch. I hiked the
Camino Frances (SJPdeP to Finisterre) from 1st Sept into early October and also needed my woolie hat and neck buff too on occasion (especially if we left before dawn to beat the heat). Not sure how cold it gets between Logrono to Burgos your time of year but I don't expect it'd be too bad. So long as you have a good outer layer (rain jacket) to keep you dry and protect you from cooler breezy weather you should be grand. One thing I did notice is that when the weather turned cooler/wetter my spare clothes were not always dry the next day. So just bear that in mind when packing. 3 mid layers (tshirt and long sleeve) plus fleece and rain jacket should suffice. Just layer up when cooler.
Also as you get closer to the trip check the long range forecast online for the towns you will be passing through. It will give you a rough idea of the weather and (lowest/highest) temperatures you should expect when you are over there. This is a great website:
http://www.accuweather.com/en/es/logrono/308463/october-weather/308463
Hope that helps!