Psyiologically, it is probably the other way around. If your brain senses that your core is in danger of not staying warm enough, it will divert more blood to do just that, which leaves your extremities vulnerable to the cold from lack of blood flow. The gospel on cool or cold weather activity is always to keep the core protected, and use breathable layers to keep the sweat from cooling you even further. Something like a wool base layer can be all that it takes to tip the gear that you have already been using to the point of being completely adequate.
You also want to keep your head warm, as that is the other large area of core heat loss. Fleece caps work a lot better than helmet covers at keeping your head warm. Helmet covers are good for blocking wind.
If it is really cold riding, adding a windblocking layer, or wearing something like pi amphip bibs that integrate a wind blocking fabric design can make a big difference. Same goes for gloves.
I like balaclavas for winter, in varying weights depending upon how cold it is. For feet I like DeFeet Inferno socks over a thin wool liner sock, again depending upon how cold it is. Wind blocking shoe covers can be added over that as required.
And anybody who thinks you can't stay dry in the rain hasn't used Showers Pass gear properly. It is as close to bulletproof as you can get, especially if you use the full hood and use it properly. In horizontal rain, adding a neck gator can be all that it takes to stay really dry inside.
Commuters rock. 😎