Went out for a ride yesterday. This is my first season of riding in snow and ice. 40 mm Studded tires on a beater hybrid is working out fairly well, but......

The paths where I ride get enough foot traffic to trample down the snow. Riding over the freshly trampled snow is a bit jarring, and both the front and rear wheel can feel squishy and slide around a bit. I am sure that once the trampled snow thaws and freezes the squishy feeling will be replaced with ice pure jarring. This dramatically increases the workload, cuts the speed, and takes from the pleasure of the ride. On stretches where there is enough untrampled snow to the side, the bike just smootly cuts through the snow and feels great.

So, is this as good as winter snow riding gets, or can I modify my bike to deal with the trampled down snow better? Front suspension, wider tires, thinner tires? Not interested in street riding in the ice and snow - I have no faith in the auto drivers on slippery surfaces.