Horses are pretty resistant to the cold, but if he wears a blanket in his stall and then suddenly finds himself “naked” as you begin to tack up, it can be a bit of a shock to his system. - Don't make him feel Naked and alone!
A kinder way is to leave his blanket on while you gather up everything you need - grooming kit, tack, leg protection, helmet and whip, etc. Next, groom his legs, pick out his hooves and brush his mane and tail. - Still with the blanket on, put on his leg protection.
When grooming, fold the blanket back so that his front end is unblanketed, but his hindquarters are still covered. It’s fine to reverse this and leave the front end covered and the back end to be groomed. When done with whatever half you worked on, cover him back up and work on the other half.
When ready to tack up, consider switching out the blanket to a quarter sheet to keep his hindquarters covered and warm. These large muscles can take awhile to warm up, so keeping them blanked until you are working is a good idea. If you use a metal bit, take a few minutes to warm it up with your hands or warm water. Your horse will appreciate it!
Once you start riding, take lots of time to warm up your horse slowly. Walking around the arena perimeter gives your horse’s muscles time to get ready to work. The older your horse, the more warm up time he may need. In the meantime, you can do your own stretching/warm up exercises.
Once your ride is over, take the time to cool out your horse thoroughly to help prevent muscle cramps. Add a cooler to keep him from getting a chill while you do your post ride groom. Then add back his blanket to keep him comfortable in his stall.