Winter tyres: steel rims vs. 235 LMF

Hi, I bought a new car, which comes on 235/40 ZR19 summer tyres. Now I wonder how I can get the best of winter tyres. As I see it, I could either buy winter tyres in the same dimensions, which would of course be more expensive, but save the money for the rims. Or narrower, smaller winter tyres (which dimension would make sense?) — for the time being — buy on steel rims. But then at least in the long term the price of LMF added.. Which option would be more economical? Of course, I want to save the least on the tires themselves.