One of the best things you can do to prepare for training and improve mobility overall happens way before you even get to the gym
 
A morning warming routine doesn’t need to be anything complicated. Just take 5 minutes as soon as you can after getting up to run through some simple drills that get your body moving againt after being still all night.
 
What I’m doing here is my typical morning warming routine, but you can and should make changes based on what you need to focus on.
 
Some mornings I’ll add a couple other movements if something specific needs more work, or spend extra time on certain drills as needed.
 
Just cover the obvious—shoulder, trunk, hip, knee and ankle movement through a wide range of motion.
 
Don’t push hard though—this is not the time to strain, but to just loosen up and reset your baseline after a long period of doing nothing.
 
Just this few minutes of simple, easy activity right when you get up will make a big difference in how you feel in training later in the day, and will contribute to improving your mobility overall.
 
Plus… it just makes you feel better at the start of the day, and that alone is more than enough reason to do it.
 
Morning warming can be as simple as doing the same basic DROM warm-up you use before training—that’s essentially what I do.
 
Or you can do things that have a similar effect that you don’t have time to do in the gym, or are similar enough to be redundant with the rest of your warm-up, but different enough to merit including somehow.
 
Just get started and experiment day to day to find what works best for you.
 
And yes, sleep itself is a big part of training prep and mobility—if you’re chronically underslept, you’ll also be chronically tight and stiff.