I have a couple strategies for carrying a personal survival kit, known as a PSK in survival circles.
First of all, it doesn't take a lot of space or weight to be prepared for losing your pack or a major component of your kit.
As others have mentioned, a small selection of core items can be carried on a necklace or in your pocket. I think that should always be the case. IMHO, you should always have a knife, compass, whistle, fire starter, and light source on your person at all times. With that small selection of tools you can build a shelter, make a fire, get found, or get "unlost."
Some like to carry a small belt pouch with a few more items than a necklace can handle. There are ultralight ones made from silnylon or you can step up to a small zippered cordura belt pouch. Belt pouches used to be common in hiking stores but have fallen out of fashion and are hard to find. A military surplus store seems to be the best bet now. There are brands like Blackhawk that make all kinds of pouches. I have one made for smoke grenades that will carry a decent PSK on my belt.
There is a tradition of using Altoids tins for a PSK container, very much in the ultralight vein. It is amazing to see the range of items that people have managed to get into a small tin. The idea is to have something that can easily fit in a pocket so you always have it with you. I have used small plastic boxes with a silicone seal-- modern Japanese bento boxes. One of my favorite PSK projects is to use an eyeglass case-- very handy
My main hiking PSK goes in a one liter Sea to Summit roll top silnylon stuff sack. If I did need to carry it, I could make a simple cord sling. The hatband on my Tilly has several yards of braided 550 parachute cord, so I always have a reserve supply, as is my bear bag line.
I have used a drawstring pack for my bear bag, carrying my food under way, and can double as an "away from camp" day pack. These simple packs have plenty of space and can hold a day's supplies and an extra layer of clothing.
I don't care for fanny packs unless they are really small-- a liter or less. I found a used silnylon one that is very light and would make an excellent PSK container.
As to carrying redundant items, it rubs up against some canon rules of ultralight hiking, but I'm adamant that redundant fire starters and a backup compass should be carried. That amounts to a few matches, a mini firesteel, a mini Bic lighter, and a zipper pull compass. I also carry a spare one liter Platypus in my PSK with Micro Pur tablets taped to it. I have a couple kinds of cutting tools as well, with a couple singe-edged razor blades in my PSK, a Victorinox Classic on my lanyard, and I generally carry a larger knife like the Victorinox Farmer or Trekker. That is the total redundancy in all my kit and I'm more than willing to take the weigh penalty for the security added. The rest of my PSK is a space blanket, a small first aid kit, a flat pack of duct tape, a bit of braided line and a few grams of fishing gear in a pill box. Nothing earth shattering-- or back breaking.