Well given that where I live it's normally 25-35C I don't have a lot of experience with snow, but if the density is consistent how about measuring the weight of the snow in a container.
The density is not consistent enough for high accuracy of snow depth, but would you need high accuracy?
presumably a rough guide to each new fall's amount is good enough. You might have issues with the
weighing mechanism freezing solid though.
There might be a way to sense snow capacitively, but I'm not sure - wet and dry snow might behave
very differently as dielectrics.