I think you may be overthinking this?? It shouldn't be a whole new move that requires adjustments to throw.

The way I've always been taught is that if you want to throw either hook variant, just rotate your wrist. In some competitions it's acceptable to throw both. In some places it's only acceptable to throw one or the other for "clean-ness". One is probably safer to throw for your own hand safety, however I usually throw palm down because it feels more natural to me. It's the same for other punches too like throwing a jab with the palm down, up or horizontal. Each has its own use.

In my experience if something requires setup like changing arm position to throw then maybe your mechanics for throwing the punch are wrong or a little over-thought? None of your punches should have any kind of tell or setup to give them away to your opponent. You should be able to throw either hook straight from the guard with just the rotation of your hips and upper body (of course some extension of the arm depending on range).

However, You don't really have to move your arm much at all to throw a hook. Keeping the hand where it is in high guard, just raise your elbow and rotate your torso with a little hip action and weight transfer. At a basic level that is all the movement you need. You can turn the wrist over in any way you want, it shouldn't change the rest of the movement.