Friends without children put their gifts under the tree as soon as it goes up: gifts for each other, gifts received and gifts for others. So, when i visit before The Big Day she rootles around under the tree and finds my gift.
I have a very hairy dog, and the thought of having to dust presents everyday and de-hair them makes me shudder! Can't be arsed with extra work. So, for us, gifts from others go under the tree Christmas Eve in the evening, our 9yo can give them a squeeze, a shake and a smell! Then other gifts go out when he's in bed.
The gifts are currently unwrapped, in a couple of bags for life in the futility room. DS hasn't gone looking for them. I've told DH where they are so he doesn't accidently tip them out and use the bags for the weekly shop.
I have a huge pile of fabric bags and hemmed bolts of fabric, all in different sizes, with lots of ribbon and skinny rope, so wrapping will take 30 minutes max. We don't warp stocking gifts, the stocking is the wrapping.
I find good hiding/storage places to be:
In a bag hanging off a coathanger in the wardrobe.
At the back of a high cupboard in the kitchen that we use for things we don't often need (that super-sized salad bowl for dinner parties, the sandwich maker... etc).
In the sock drawer
ain't no one going in there!
Behind books on teh bookcase (I have an Ikea Kallax bookcase that holds my books three-deep, i can move books around and create hidden spaces at the back).
At work (I don't WFH).
On top of the fridge behind my mixing bowls (nothing edible as it can get hot up there).
Anything random, like empty cereal boxes (they'll obviously feel heavy when you move them, so shouldn't be chucked out accidentally).
We move a lot, and I've kept boxes for things such as my sewing machine, blender etc that i used to re-pack, gifts can be stored in them.
Anything small can be put in a sandwich bag (to protect against dust and humidity) and slid under the sofa, or the bed, even if these are quite low to the ground.
I hope some of these help.
Just don't do what one MNer (or her partner, don't remember which) did one year: stored in the shed in binbags, threw them all out 