First, how cold is cold where you intend to ride? There's winter in SoCal, and winter in Alberta.
Second, in winter are you a dry, sunny day cyclist, or an all-conditions rider?
Finally, everyone's tolerance for cold is different. What's yours?
That said, road shoes are designed to keep your feet cool. There's little that can be done make them warm. In my experience, neoprene booties are effective only to the mid-30s. Below that, one way or another, I need different shoes.
If I were to go the lots of woolly socks route, I'd need bigger shoes to fit them in. Since we were talking new shoes anyway, I chose the winter cycling shoe route. I'm an all-conditions commuter, so I ride SPD, and chose the Lake MXZ302. They get me down to around 10°F with just one pair of plain athletic socks.
The same shoe comes in a road model for three-bolt cleats, the CXZ302
. Several sunny day winter cyclists in my club have them. When we ride together, we look like an advertisement for Lake. Those that don't own Lakes, own the Sidi winter cycling shoes. Those that don't own either, stay home.
Lake now makes a cheaper winter road shoe, the CX140
. I don't know anyone who owns them, so I can't report on the differences.
There's absolutely no reason why you should have to switch to platforms if you don't want to. Just as there's absolutely no reason why you should ride clipless if you don't want to.
But if you want warm feet, you'll need new shoes. Either larger ones so you can fit woolly socks inside, or winter ones.
EDIT: I use the word "shoe" generically. It includes sandals. It's beyond my comprehension, but there are many winter riders here who swear by sandals and woolly socks.