The best solution I found was a wire live trap, a long pole, and a 50 gallon barrel filled with water.
Gently lift the live trap with skunk on the long pole and gently lower into the barrel of water. The skunk will drown most of the time without releasing any of the musk. Even if they do, all you will see is an oily slick with no odor, but they rarely relesed any musk.
I will be surprised if you only have one skunk; chances are it is a colony or a pregnant female, or thats the way it usually turned out for me.
Good luck and move slow.
P.S. Handled them different in Coyote trap sets; would lash my very sharp pocket knife to a long pole, and with the skunk facing you gently find the opening into the chest and gently thrust the knife in. Sticking the knife in isn't the problem. If you can wait a while the skunk will bleed to death internally.
Most of the time, if you are in a hurry (and what trapper isn't always in a hurry?) you can gently slide the knife out! It must hurt on occasion, and when it does the skunks lets loose and empties the musk glands.
Which creates the mess you are trying to avoid, but it will enhance the set for Coyotes. Just blind set the high spot on the edge of the circle. The next Coyote will be smelly though!
Just my experiences.