I’ve spent much of my working life having to make notes on big, flappy plans or small, hand-held notebooks, so this comes with a degree of personal experience.
Consider a slightly-larger-than-tiny memo book?
Learn to tuck the book-holding arm against your body to stop that hand ‘moving around and getting pushed back’, but also think about angling that notebook in whichever way makes writing easiest, maybe with the bottom of the page angled with its left corner towards you and the right at 45°. Experiment, but don’t assume that flat and square like a desk is best. I find that tucking that bottom left corner of the book against my body helps to brace it.
Try a fibre-tip pen rather than a pencil so that you don’t have to press as hard on the point. Or get a very soft pencil.
Don’t try and write while you walk, step aside out of the footway so you don’t get jostled.
Try to learn shorthand or devise your own version. That would limit the amount of writing.
Practice handwriting in a relaxed way; much of the movement should come from your shoulder and upper arm rather than wrist and fingers. If you can master that, having little space to rest your hand will matter less. This blog has some good explanations and exercises. Eg
To get a feel for the proper muscles (and start training them correctly), hold your arm out in front of you, elbow bent, and write in the air. Write big. Use your arm and shoulder to shape letters; hold your forearm, wrist and fingers stationary and in writing position. You’ll feel your shoulder, arm, chest and some back muscles doing most of the work. That’s good. That’s what they’re supposed to do. Try to duplicate it each time you practice.
Write in the air until it becomes as natural as breathing. It’ll be awkward and feel silly at first.
Above all, remember that your notes only need to be legible to you, you don’t need to win handwriting competitions with them. But make it a habit to transcribe the notes as soon as possible, while you still remember what your scribbles meant!