The idea is to keep the poultice from drying out so you want to seal in away from the drying effects of the air. Ditto with the hoof itself, as keeping it softer helps provide an easier path for the abscess to travel out. With thin soles, you need to balance the need for softening with the need not to compromise the sole any more than it already is.
Plastic wrap will keep everything from drying out and it's readily available and inexpensive. You can cover the coronary band with cotton soaked in epsom salt solution, then plastic wrap over the top of that to keep everything moist. A diaper over the top of everything will insulate, pad and absorb anything that drains out. Then stick it in a boot if you have on that fits - or just use vet wrap and duct tape.
To soak, you can use a soaking boot: pull the boot on, pour the solution in, cinch/duct tape it closed around the pastern and let the horse wander around or put him in a stall with hay to distract him. I've used the Clean Trax boots to soak - they are a tall boot that goes up over the knee, with a nylon cinch strap at the top. Put the boot(s) on empty, then add the soaking solution. Then I attach a lead line to the strap, run it over the withers and attach it to the other side of the boot strap. This keeps the boot from sliding down without you having to watch it constantly, plus it allows the horse to move around a bit instead of having to stand still in one spot.
I wouldn't recommend casting an abscessing foot as that makes it harder to access it for treatment. Plus, an abscess is going to follow the path of least resistance and what that path is depends on where the thing is located inside the foot. Gentle movement, which will mechanically create expansion and contraction inside the foot will help get it moving. Jiaogulan, which helps increase circulation, can also help encourage abscesses to come to a head.
Don't know that there are any statistics on abscesses resolving without draining out. Some may drain out around the frog and not get noticed at all.
NSAID use will slow maturation of an abscess so should only be considered if the horse is so sore it won't eat/drink well.