I used to use a Michaels of Oregon (Uncle Mike’s) belt pouch to contain an S&W J-snub, which I hand-carried as I ran laps around a wooded park. To equalize the arm workout, I would switch hands each lap. Another option, for an out-and-back run, at other locations, I would switch hands at the turn-around point. This pouch is no longer available, but, there are plenty of similar-sized nylon fabric pouches on the market. This was a zippered pouch, made to wear on a belt, not a “fanny pack” or sling bag. If I remember correctly, I eventually started using a length of para cord, as a “dummy cord,” in case I were to stumble and drop the bag.
One could opt to carry a balanced load, in the other hand, rather than swap the one bag, back and forth.
A small sling bag could be hand-carried, similarly. The strap could pass over one shoulder, but instead of the bag being allowed to bounce, freely, a hand supports its weight. I have not remembered the brand name of the bags I have used, in this manner.
I no longer run hard enough to need a special set-up, while walking or scrambling about, because I do not induce enough of a “bounce” for long enough to be a bother. Any running I now do is breaking into a run, for short periods, while walking.
Typing this makes me feel old.