All righty, I've got an update.
I called several local tire stores, and unfortunately, the General Altimax HP is being discontinued. No one knows exactly what it's being replaced with, but they suspect it's General Altimax RT. But since these tires are so new, no one was able to give me adequate feedback about them. I tried finding Continental DWS and was told that they're not available in my size.
So, I asked for recommendations. At the first store, I was told that Firestone Precision Sport is a good tire for my needs, but after looking at the photo, I wasn't convinced. I called the same store again, got a different salesperson, and was told that the Firestones are a bit of a step up from the Hankooks, but not enough to justify the cost. They said I'd be better off just getting winter tires. They recommended Blizzaks.
I decided to call another store to see what they would say. The salesperson recommended 3 options for all weather tires: Goodyear Assurance Triple Tread (with a rebate, I'd be looking at about $630 for a set of 4 (quite a bit above my budget, frankly); Goodrich Advantage TA (they're about $109 per tire, so would end up $519 plus tax); and Yokohama Avid Ascend (these would run me $417 plus tax, but she honestly said that, while better than the Hankooks, they may not be as big a step up for winter as I might want). I decided to spend tonight looking at reviews and seeing what you guys think. Honestly, at this point, and based on the reviews I've seen, I'm leaning towards Goodridge Advantage TA. Many people comment on the great snow and wet weather traction, and the price is right. The Goodyears also get amazing reviews and frankly they're probably somewhat better in the snow, but the price is super steep, and I wonder how they perform in warm and dry weather.
PS, all quotes include free rotations and other maintenance / 30 day guarantee related stuff you usually get at places like Sullivan Tire.