Since this is about work, I'll share my "At Work" methods.
A few things to mention. I work in a factory, no one has any clue I wear padding, I have access to my locker at any time because it's right on the main work floor and I do not have to share it with anyone.
I used to keep a change in my locker in a small grocery bag, but it was still a hassle to get it out and get to the bathroom. I needed a better solution. I usually carry a LOT of small, but needed, items* in my shirt pockets. Trouble is, I have to bend down to pick things up and out everything would come. So I got a small string backpack, the kind that just cinches closed and the strings double as the straps. I put all of my small items in it, my needed snacks and my extra padding.
No one knows and I doubt anyone cares. I carry the pack with me everywhere and no one has said anything yet. If they do, I WILL bring up the need to HR. When I am working on equipment, I take it off and hang it up nearby. If I leave the area, I take it with me.
Only one person has joked about seeing what I have inside my bag and I said "It would be both glorious and horrifying." And that was it.
So, if this might work for you, give it a try. After people seeing me with the bag all the time, no one even notices when I take it in the bathroom. Once business in the bathroom is concluded, I just wrap the used padding up and put it in the bin. I do NOT try to cram it into those little 'sanitary' bins for tampons/pads. That would be horrible trying to get it back out and I don't want anyone to get that familiar with my pee.
(For those curious folks: My bag contains a small plastic figurine of Rocky from Paw Patrol for good luck, a stuffed sweet-smelling pink ram, two ink pens, a tire-pressure gauge, my wallet, work ID, keys, MP3 Player, phone, padding, a bag of nuts, a bag of jerky and a packet of crackers.)
