Fortunately for me, I guess, I get to practice wet firing lighting here in Oregon three quarters of the year.
The main difference between lighting fires in ideal conditions versus damp, (when not "cheating" with man made accelerants), in my experience is that in wet conditions there are no shortcuts. You have to carefully prepare the site and have everything preassembled to "always" succeed.
Tips:
Carefully select your spot. When it is raining or foggy, I make a fire circle in the rain shadow of the largest evergreen in the area. A modest fire about five feet from the massive trunk does three things, two of which are positives, 1) the ground is way drier. 2) The trunk can be the easy basis for a heat reflector. 3) Watch the breeze. A canopy that is keeping the rain off of you also traps a lot of smoke if your fire can't or won't burn hot enough to keep the problem in check. ON EDIT: If you build a fire under a tree, clear a fire circle of all of the grounded needles, twigs, and cones unless you want a spontaneous combustion problem from an ember and keep the fire size sensible.
Procure the driest tinder you can. I have it easy here in Oregon. There is usually tons of fuel available on the ground or within easy reach. Moss is everywhere and locating some dry moss is usually not tremendously difficult. Rotting logs sporting termite colonies will have exploitable termite crap that burns nicely. Finally, exploiting deadfall is key for firewood. Even a branch or small tree that has been rained upon for weeks on end has a drier side towards the ground. I usually break it, relocate the "dry" side, split it open and fuzz up the interior. In moist conditions the tinder is better when the curls are thin and springy. IMO the tinder pictured above was too big and not adequately assembled for airspace to succeed in wet conditions
However, another method is even easier if there are mature cedar trees in the area. Most cedars of intermediate size will have failed branches still sticking off of the trunk that died off for lack of sunlight. These are easily knocked to the ground with a stick. Since they have been rain shadowed, they snap easily and burn nicely. You only need one if you process it into shavings, bark and all, to get a hotter burn for moister deadfall fuel.
Buy or harvest some fat wood or look for tree damage and trim off some pitch balls. In damp conditions having some pitch or pitchy wood to catch is the easiest fix for building a fire that won't fail in moist conditions. I usually "cheat" on this count by always carrying hexamine tablets, but I only rarely have to use them. I just find hexamine cleaner to carry and use than pitch and less messy than cotton balls in petroleum jelly.
So, a dry fire site, dry tinder and fuel, and useable accelerants can all be found in a temperate forest, even when everything looks wet.
If you're not in such a forest, good luck. Someone else will have to chime in.
Slip a couple birthday candles or a bit of wax into your pack. A few slivers of paraphin burn for a shockingly long time, and can help dry out your starter wood.
