well, nice interesting topic and good question, that I think those of us that do it take it a little for granted, for those who run "busier-cellphone infested" lives. As one who practiced martial arts for 30 years and what I learned in the Marine Corps.
When I stalk this is what I do. First, take watch/cell and through time away. Your only limit is legal shooting time. If you cann't do this, then do not stalk. You may think, I have a few hours, I will stalk through this woods, then you are looking at clock not game, and that trophy will purposely move just enough to eat up 4 hours of your time.
Next, take everything in. Move 10 yards into "woods", lean against a tree, and take everything in, feel the direction of wind, what animals are chirping, listen to sound of woods. You will be suprised at once you are tuned in, and the sound changes it will alert you, more.
three, carry bino's and scrutinize everything ahead of you, only move when you know it is clear, not maybe. It is amazing if we listen to our predatorial instinct how much it works. Listen to your gut, when you noticed movement, or that just doesnt seem right.
four, do not move if you have experienced the above. Find out what it is, wait and study it out. do not let your mind say "nothing", wait to verify what it is. Could be a deer, could be a coyote tracking a deer, maybe a racoon (during day sign storm coming). Know what it around you, not speculate.
five, when you move, plan it (route selection). Do not walk like on street, lift your leg, lead with ball of foot, not heel. When you first do this, you will get tired and bored. Once, plan where your foot will go, move slow and steady, plant, slowly add weight, then move the next leg. Do not move more than three steps, then scan your area again (be amazed at what world 3 steps can open). Move as if you are drawing your bow, at game. Slow and deliberate.
six, know the wind and use it.
seven, Once you find game, good job. after taking 5 seconds to celebrate, start to plan your approach, study the route and find limits. Only move when you can. Plan on rest of day if needed.
eight, the whole time just relax, listen and feel the woods, take it all in. This is suppose to be fun and relaxing. Slow down, see a stump/down tree, sit down and relax.