I use a Lush shampoo bar and a Lush conditioner bar when I travel. I cut the bars into 1/4 or even smaller depending on how long the trip is. I take the bars out and leave them on the side of the tub/shower. When it’s time to pack for the next location I wrap the bar in a piece of plastic wrap/Saran wrap/cling film, put it in a plastic container I got from the discount store and that’s that. And yes, I reuse the same piece of plastic wrap.
Edit to add: To minimize mush, don't use the bar directly under the shower head. Thoroughly wet your hair and hands, step/lean away from the shower head, roll the bar in your hands a few times or swipe the bar through your hair, put the bar aside and work the shampoo or conditioner through your hair. Try to let the bar dry out before you pack it back up. Try not to leave it in a dish or other container sitting in water. Best if you can put the bars on a face cloth outside the shower between uses.