Help me pick some A/S tires

[ShiningArcanine]

I have Pirelli P400 tires on my Toyota Avalon and they have been on there for at least 40000 miles. I like them and they work well in the winters in New York. The only traction problem I have had with them was once when I made a quick left turn on to a road that was very icy and which also had a dip in the road; the car almost went into a spin, but I turned the wheel to the right very quickly, stepped on it and my car pulled out of the near spin without a problem. I do not think that the near spin was the Pirelli P400 tires' fault given the circumstances, but I think some of my ability to pull out of the spin could be attributed to them (with the majority being attributed to my Avalon's front wheel drive and power steering).