For people who are not very active this recently published essay might be of great help for your handstand practice and your understanding of the handstand as a skill:
https://www.gymnasticbodies.com/3-handstand-cues-beginners/
In short the essay describes 3 very important cues; "grip the floor", "shrug your shoulders" and "squeeze your butt".
"Gripping the floor" will greatly help the with the wrist-fingers re-balancing because of a larger surface of contact. Think of this when you are aiming to get beyond the wall handstands. Shrugging your shoulders or simply tensing your shoulders correctly will create a more stable and energy efficient position so that you can balance better and for a prolonged period of time. Simply press your hands as hard into the ground as possible. Squeezing your butt will prevent your handstand from arching, however if piking is your problem "sucking in the stomach" will help align you better.
These cues are very important, and you should definitively give them a read!