There is no reason why you cannot continue walking in cold weather. You must of course be adequately clothed. Heavy jackets and sweaters will make walking difficult so it is better to dress lightly but warmly. The secret is to dress in layers. By that I mean several light layers of clothing which can be taken off if you get too warm and replaced if you feel chilly. You can keep the extra layers in your backpack or tied around your waist
The first layer on top should be of a wicking material such as Coolmax which draws the sweat away from the skin. Do not wear cotton as, while cotton is very absorbent, it does not dry quickly so you will be walking in a moist tee-shirt which is conducive to catching a chill. The next layer can be a long sleeved top and over this a lightweight thermal fleece. At this stage you can use as many layers as you think necessary but be sure the outer layers are large enough to fit over the under layers and that you are not constricted in any way.
Choose a light weight, windproof, rain proof and breathable jacket with a hood. If it is not breathable the sweat condenses on the inside and the moisture will soak into your clothing. A heavy jacket is difficult to wear and carry but with a lightweight jacket you can always add an extra layer underneath if necessary. Make sure the jacket fits comfortably over all your layers.
A good fitting hat is essential as we lose 70% of our body heat through our head. There are many light weight thermal hats on the market and check that you are not allergic to any of the materials, as a constantly itching head will take all the pleasure out of your walk. Use your hood in intense wind
or rain. It gives an extra layer of insulation and protection. Check that it can be secured easily and efficiently. A hood that slides down at the slightest puff of wind is a great annoyance.
A silk scarf around your neck, which is long enough to spread out over your chest, will make you feel as warm as toast!
Choose a polyester material for your winter walking pants, as it will dry quickly. A pair of panty hose, winter weight tights or long johns under your trousers will give an extra layer. Do not wear jeans walking. They do not allow enough movement and if they get wet will not dry and may cause chafing and irritation. Rain proof over- pants are excellent in case of rain. They can be put on and taken off as necessary.
Socks too, should be a good fit and of a quick drying material. Two pairs of light socks are better than one thick pair and make sure that your feet have plenty of room and are not crushed in any way. If they are it is a sure recipe for blisters.
Switch to winter weight walking boots for cold weather. They must have a long integrated tongue and be waterproof with flexible sole and good deep cleats to cope with ice or snow.
To complete your winter walking outfit a pair of warm gloves. Again two thin pairs will keep you warmer than one bulky pair. The air between them acts as insulation.
As you walk swing your arms and keep up a reasonable walking pace. If you feel chilly quicken the pace a little.
Enjoy your walk!
Here is a link for a very practical base layer wicking tee-shirt.