Stretching full body everyday vs. switching between lower and upper body?

First off, I can't wait for Stretch 1 to come out because I lack knowledge in the flexibility department. With that being said, I hear the phrase "stretch everyday!!!" all the time. Although should I give muscles a "rest day" after stretching them like how you would in strength training? Or is a full body stretching session more beneficial if performed everyday? 

I also realize there are many types of flexibility training but in this context I'm referring to standard static stretching.

-Thank you!

Some stretching everyday is beneficial especially after waking up and just after a workout as a "Cool down." You don't have to dedicate a full hour to stretching. If you spend some time web searching you can find stretches that you can do sitting at your desk in your chair at work.

 

As a warning: There does come a point in time where your stretching seemingly doesn't feel like stretching because you can't feel tension. Resist the urge to tweak your body further for the sake of finding tension. Yoga practitioners injure themselves when they start doing this.

 

When you have reached a point where you don't feel tension you are at the point where your stretching becomes maintenance.

I don't think it is necessary to give your muscles "rest day" for stretching. I am currently doing flexibility training ~1.5-2 hours a day, 7 days a week. I think the results are absolutely phenomenal. After 4 days of training, I went from hanging my arm ~30cm above the floor to be able to touch my toes (just barely though). You can grab a copy of Thomas Kurz' Stretching Scientifically and work on that. I recommend the isometric stretching section.

What kind of stretching do you require at a daily basis after waking up? Something very quick, for I dont have that much time

I've always found that Daily mobility and full body light stretch combined with 1-2x week of Hard stretching (trying to push it to the limit) is a good approach.  Also taking a rest every now and then from hard stretching seems to help myself and a lot of people.  I know at the circus school, most people (except the contortionists) would come back more flexible after the holidays from no stretching. I can't say 100% that that was because of a brief period of rest from stretching, but it was pretty consistent. Because of this, I ways periodize my stretching just like strength work (having deload weeks 1-2 month).

 

Daily mobility would be moving and 'greasing' up all the joints. Arm/wrist/shoulder circles etc, + Thoracic mobility on the roller and opening the hips etc.

 

Hope that helps