It says you can't move while attached, it doesn't say you can't move to attach. Just one bodypart needing to be still relative to [object].
How would you define stop? If I jump, there's a very short moment at the peak where I'm not falling, if I put my hand on a car, it rests there the moment I stop giving movement impulses.
Attaching is a simple action, which means, taking a non-combat adept with ~2 Ini passes, that it logically has to happen in under 1 second.
You can easily attach a hand to a surface by just grabbing slightly below your max, for example if running up the wall (without adept powers, just oldfashioned momentum), by the time you lose grip on the wall with your feet, put your hands on the wall, then the momentum carries you slightly upwards still, but by the time you fall your magical energy should have connected your hands to said wall and you will stick.
As you're not actually moving your hands on the wall, no problem, the arms attached to that hand don't count. You won't be able to continue upwards at a later date on a plain wall, however.
It also nowhere says that you have to attach a lot of bodyparts, just one will do; there's a clear or there.
Also, hang time provides absolutely no bonus to climbing as climbing is a repeated complex action and not an extended test; per RAW, you never actually tire.