In addition to riding a bit farther from parked cars, get in the habit of watching for heads in those cars. Easy to see once you start looking. Every occupied car is a potential door-opening threat.

As for tire choice, it can't hurt to look for a softer compound, wider tire, larger contact patch, and some tread pattern. I disagree that a narrow, high-pressure, slick is as grippy as a wide, lower-pressure, treaded tire in real world rain commute conditions. In the winter, streets are not only wet, they have a lot of debris. Suppose a narrow, high-pressure, slick tire rolls over a piece of debris - a very small pebble or a bit of twig. The debris blocks a large portion of the small contact patch. The high pressure means the tire deforms less, and the remaining contact patch is more likely to be lifted off the road surface. The slick surface also means there is no channel or sipe for the debris to sink into, which also tends to lift the contact patch. Now suppose a wide, lower-pressure, treaded tire rolls over the same debris. The debris blocks a small portion of the large contact patch, the softer tire deforms more easily around the debris, the debris also has a chance to sink into the tread pattern. Take it to an extreme and imagine rolling over a pebble in a 19 mm 170 psi tire versus a 40 mm 60 psi tire.

The other factor is tire compound. A softer compound will give more grip at the cost of shorter life and higher rolling resistance. So no harm in looking for a softer compound tire as well.

That said, no-one is going to put a 40 mm tire on his bike that currently wears a 19 mm tire, just for better rain traction. The actual choices are a lot more subtle - hmm, should I get these 23 mm tires or these 25 mm tires? So I suspect the amount of difference that this all makes is also rather subtle.

Personally, I commute year round in rainy Portland on 25 mm Panaracer Paselas which have tread but aren't advertised as a "rain tire", at about 100-120 psi, and they are perfectly fine.