Carry extra rounds? Horse pucky! That's why we hunt with these single-shots... we don't need more than one good shot.
It's a personal preference thing, so YMMV from mine!!!
I guess I've tried just about every way possible to carry extra cartriges for the TC's over the last 42 years to hunt big game, small game, upland, waterfowl, predators and varmints. Some types of hunting require different carriers than others.
In a big game hunting situation some carriers work well, some do not. First up is to decide on how many extra you want to carry. For quick follow shoots 2 extra cartriges should be plenty IMO, and they should be carried differently than your back up supply of ammo for the day. I think your "quick" carry should be 3 rounds so one round can be used to easily load the firearm at the "start" of a hunt and as a place for it to be returned to at the "end" of it.
A backup supply is a good idea, but If you think you need to carry a full box though you're using the wrong gun - get one you can hit something with.
Your ammo wallet is great for carrying usually 5-12 cartridges for an all day hunt away from the camp/vehicle. I have several and use them for just that purpose. But they are not all that fast to use for a quick followup shot when that 7 point royal needs one more to go down. The belt slides are faster, but still have the problem of being burried under layers of clothing at the worst possible moment. Sling loops might be a little hard to find by braille when you're eyes are glued to your elk, especially if the sliing is flopping around. IMO those wrist carriers are the most worthless of all the ways to carry. What worked for me best was a stock carrier. Not those cheap slip-on thingies with elastic loops that hold 6+ cartridges (to make up for the few you'll loose out of it during a day of hard hiking in the thick stuff). Get one that laces on, preferably leather with tight leather loops and that only carries your 3 cartridges. I made all of mine myself, easy. With them the carts were easy to find by braille with my off hand and without loosing my grip on the rifle with my shooting hand.
Loading for a fast follow requires pratice, especially to do it without taking your eyes of your game animal. So it needs to be practiced same as your shooting.
Since most of my big game hunting with a Contender was with pistols, I choose to live with a sling carrier some of the time, and with the leather belt slides some of the time and just made sure they were always were they could be accessed easily with my left hand. Both worked good enough, but they were not as fast as the stock carriers were.
L.