Our body's natural reaction in winter is to become lethargic + hibernate + conserve energy. Some of us experience seasonal fatigue (Seasonal Affective Disorder) that results in depression, tiredness and constant hunger. Also, our muscles are more tight than warm and loose. Tight muscles lead to tight joints.
Whatever time you typically spend stretching, increase it by 2X
If you are stretching outdoor, make it an active dynamic stretch. The last thing you want to do is sit down relaxed in 40 degree weather.
Use a stretching device if need be ( cord, foam roller, strap ) . Lately I've been using yoga blocks. These props are not only meant for beginners. You can use them in different ways to improve your stretching routine and flexibility.
On a non-winter related note, for those wanting to improve their flexibility, its important to practice daily. When you stretch, its okay to extend to a slight uncomfortable level. However, understand the difference between subtle discomfort and point blank pain. It takes time and consistency to improve your flexibility. And even then, some days you'll find your muscles too tight and unwilling to corporate. The key is to listen and respect your body. And to never "rush" it to a limit its not ready for.
Don't forget to B.R.E.A.T.H through your routine.