I think in a lot of Italy, October is still considered on the high season side and many travelers are still out and about. I went to Italy in the Autumn too, and it was still very warm and balmy.
My experience for a later time of year -
When I went to Rome two times, it was in December. It was a comfortable crowd at some of the major attractions. Future trips to Rome will most likely be in December too. The weather was quite pleasant - first year had a brisk weather atmosphere; windy but not too crazy. It rained a bit the morning I visited The Vatican. Thereafter, "perfect."
Second year, there was an increase in temp - a couple of days saw 70 degrees. One day it rained a little and turned muggy. People carrying their coats, etc. At night, it went down to "near normal temps." The rest of the days were beautiful.
Regarding St. Peter's - Most of the time, when I went, there was no wait - walked right up to the security center, and in I went. There was a time or two it seemed I had the whole Basilica to myself. A fraction of the people compared to high season I am sure.
Still, Rome had its moments then.
December 8 (A National Holiday) and December 12 (A celebration Mass of Our Lady of Guadalupe) are two important dates in Rome. Both were quite crowded; falling on or near a weekend.
On December 8th, the Feast of The Immaculate Conception is celebrated first at St. Peter's Piazza - Pope Francis gives his Blessing, hops into his Pope mobile and heads down to Piazza di Spagna (just as crowded) for the celebration and prayers to honor Mary.
Pope Francis lays a laurel wreath at the foot of the column with prayers. (December 8th is the official start to the Christmas season - The Nativity and Tree is up at Piazza di San Pietro.
If a person is closer to the Spanish Steps, they would need to be at Piazza di Spagna very early to get in the front row. So, a choice would need to be made - either at St. Peter's or Piazza di Spagna to see the Pope.
If you ever visit and get to see him, he is of a gentle spirit; has a shining aura about him. Of course he has his security detail with him. But, he has stopped to bless a child; make a sign of the Cross to the crowd. He prefers to not have his hand kissed. A big no no with him.